problem solving with factors

Factoring is an essential part of any mathematical toolbox. To factor, or to break an expression into factors, is to write the expression (often an integer or polynomial ) as a product of different terms. This often allows one to find information about an expression that was not otherwise obvious.

• 1 Differences and Sums of Powers
• 2 Vieta's/Newton Factorizations
• 3 Other Useful Factorizations
• 4 Practice Problems
• 5 Other Resources

Differences and Sums of Powers

Using the formula for the sum of a geometric sequence , it's easy to derive the general formula for difference of powers:

This also leads to the formula for the sum of cubes,

Vieta's/Newton Factorizations

These factorizations are useful for problems that could otherwise be solved by Newton sums or problems that give a polynomial and ask a question about the roots. Combined with Vieta's formulas , these are excellent factorizations that show up everywhere.

Other Useful Factorizations

• Binomial theorem
• Simon's Favorite Factoring Trick
• Sophie Germain Identity
• Factor Theorem

Practice Problems

Other Resources

• More Common Factorizations .

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Factors, Multiple and Primes - Short Problems

Producing Zeros

Weekly Problem 10 - 2008 If the numbers 1 to 10 are all multiplied together, how many zeros are at the end of the answer?

Factor Trio

Which of the numbers shown is the product of exactly 3 distinct prime factors?

Reversible Primes

How many two-digit primes are there between 10 and 99 which are also prime when reversed?

Multiple Years

Weekly Problem 18 - 2016 The year 2010 is one in which the sum of the digits is a factor of the year itself. What is the next year that has the same property?

Product 100

The product of four different positive integers is 100. What is the sum of these four integers?

The numbers 72, 8, 24, 10, 5, 45, 36, 15 are grouped in pairs so that each pair has the same product. Which number is paired with 10?

Tricky Customer

Charlie doesn't want his new house number to be divisible by 3 or 5. How many choices of house does he have?

Calculation 2000

Weekly Problem 49 - 2013 What is the value of 2000 + 1999 × 2000?

Stamp Collecting

Last week, Tom and Sophie bought some stamps and they spent exactly £10. Can you work out how many stamps they bought?

Prime Order

Weekly Problem 24 - 2006 How many of the rearrangements of the digits 1, 3 and 5 give prime numbers?

Almost a Million

Weekly Problem 26 - 2014 Which of the given numbers is divisible by 6?

Divisible Digits

Can you find the missing digits, given that the number is divisible by 3, 4, 5 and 6?

Little Goldbach

How many of the numbers 1 to 20 are not the sum of two primes?

What's on the Back?

Four cards have a number on one side and a phrase on the back. On each card, the number does not have the property described on the back. What do the four cards have on them?

Multiplication Table Puzzle

In the multiplication table on the right, only some of the numbers have been given. What is the value of A+B+C+D+E?

A whole number less than 100 gives remainders of 2, 3 and 4 when divided by 3, 4 and 5. What is the remainder when it is divided by 7?

Grandma's Cake

What is the smallest number of pieces grandma should cut her cake into to guarantee each grandchild gets the same amount of cake and none is left over?

Cakes and Buns

Helen buys some cakes and some buns for her party. Can you work out how many of each she buys?

Find from Factors

Weekly Problem 35 - 2006 A number has exactly eight factors, two of which are 21 and 35. What is the number?

Jenny's Logic

Weekly Problem 52 - 2009 How did Jenny figure out that Tom's cards added to an even number?

Back of the Queue

Weekly Problem 48 - 2013 What is the remainder when the number 743589×301647 is divided by 5?

Digital Division

How many three digit numbers formed with three different digits from 0, 1, 2, 3 and 5 are divisible by 6?

Smallest Abundant Number

Weekly Problem 34 - 2017 An abundant number is a positive integer N such that the sum of the factors of N is larger than 2N. What is the smallest abundant number?

The five digit number A679B, in base ten, is divisible by 72. What are the values of A and B?

Adjacent Factors

Two numbers can be placed adjacent if one of them divides the other. Using only $1,...,10$, can you write the longest such list?

Ones, Twos and Threes

Each digit of a positive integer is 1, 2 or 3, and each of these occurs at least twice. What is the smallest such integer that is not divisible by 2 or 3?

Halloween Day

One year there were exactly four Tuesdays and four Fridays in October. On what day of the week was Halloween.

Red Card Blue Card

Can you arrange the red and blue cards so that the rules are all followed?

Cinema Costs

Weekly Problem 41 - 2009 At a cinema a child's ticket costs £4.20 and an adult's ticket costs £7.70. How much did is cost this group of adults and children to see a film?

Seven from Nine

In how many ways can seven of the numbers 1-9 be chosen such that they add up to a multiple of 3?

Colossal Sum

What is the units digit in this sum of powers of 9?

HCF Expression

Find out which two distinct primes less than $7$ will give the largest highest common factor of these two expressions.

Powerful Finale

Weekly Problem 24 - 2017 What is the last digit of $3^{2011}$?

Given any positive integer n, Paul adds together the factors of n, apart from n itself. Which of the numbers 1, 3, 5, 7 and 9 can never be Paul's answer?

Leap Monday

Can you find the next time that the 29th of February will fall on a Monday?

Missing Digit

What digit must replace the star to make the number a multiple of 11?

Trailing Zeros

How many zeros does 50! have at the end?

Four or Five

The diagram shows a large rectangle composed of 9 smaller rectangles. If each of these rectangles has integer sides, what could the area of the large rectangle be?

Big Blackboard

Can you work out which numbers between 1 and 2016 have exactly two of 2, 3, 4 as factors?

17s and 23s

Can you form this 2010-digit number...

Common Remainder

144 divided by n leaves a remainder of 11. 220 divided by n also leaves a remainder of 11. What is n?

Peter's Primes

Peter wrote a list of all the numbers that can be formed by changing one digit of the number 200. How many of Peter's numbers are prime?

Indivisible

Each time a class lines up in different sized groups, a different number of people are left over. How large can the class be?

Flora the Florist

Flora has roses in three colours. What is the greatest number of identical bunches she can make, using all the flowers?

A male punky fish has 9 stripes and a female punky fish has 8 stripes. I count 86 stripes on the fish in my tank. What is the ratio of male fish to female fish?

End of a Prime

I made a list of every number that is the units digit of at least one prime number. How many digits appear in the list?

Threes and Fours

What is the smallest integer where every digit is a 3 or a 4 and it is divisible by both 3 and 4?

Essential Supplies

Chocolate bars come in boxes of 5 or boxes of 12. How many boxes do you need to have exactly 2005 chocolate bars?

Powerful Zeros

How many zeros are there at the end of $3^4 \times 4^5 \times 5^6$?

Weekly Problem 47 - 2017 How many numbers do I need in a list to have two squares, two primes and two cubes?

Divisible Palindrome

What is the sum of the digits of the largest 4-digit palindromic number which is divisible by 15?

Triangular Algebra

Weekly Problem 26 - 2017 The angles in the triangle are shown in the diagram in terms of x and y. If x and y are positive integers, what is the value of x+y?

Relative Time

Albert Einstein is experimenting with two unusual clocks. At what time do they next agree?

Times and Square

Roger multiplies two consecutive integers and squares the result. Can you find the last two digits of his new number?

Coin Collection

When coins are put into piles of six 3 remain and in piles of eight 7 remain. How many remain when they are put into piles of 24?

Added Power

How many integers $n$ are there for which $n$ and $n^3+3$ are both prime?

Sticky Fingers

Ruth wants to puts stickers on the cuboid she has made from little cubes. Will she have any stickers left over?

Eight Factors Only

We are given two factors of a number with eight factors. Can you work out the other factors of the number?

One of these numbers is the largest of nine consecutive positive integers whose sum is a perfect square. Which one is it?

Last-but-one

What is the last-but-one digit of 99! ?

Primes and Six

Weekly Problem 1 - 2015 If $p$ and $q$ are prime numbers greater than $3$ and $q=p+2$, prove that $pq+1$ is divisible by $36$.

Adding a Square to a Cube

If you take a number and add its square to its cube, how often will you get a perfect square?

Cancelling Fractions

Can you find a fraction with the following properties?

Factor List

Tina has chosen a number and has noticed something about its factors. What number could she have chosen? Are there multiple possibilities?

Three Primes

Weekly Problem 6 - 2010 Can you find three primes such that their product is exactly five times their sum? Do you think you have found all possibilities?

Fortunate Inflation

The price of an item in pounds and pence is increased by 4%. The new price is an integer number of pounds. Can you find it?

Weekly Problem 26 - 2008 If $n$ is a positive integer, how many different values for the remainder are obtained when $n^2$ is divided by $n+4$?

Using the hcf and lcf of the numerators, can you deduce which of these fractions are square numbers?

Rational Integer

Weekly Problem 39 - 2012 For how many values of $n$ are both $n$ and $\frac{n+3}{n−1}$ integers?

Factorised Factorial

Weekly Problem 17 - 2010 The value of the factorial $n!$ is written in a different way. Can you work what $n$ must be?

Super Computer

What is the units digit of the given expression?

Square Product

Weekly Problem 10 - 2011 Will this product give a perfect square?

Integer Indices

From this sum of powers, can you find the sum of the indices?

Factors And Multiples

Related Topics: More Lessons for Arithmetic Math Worksheets

If a is divisible by b , then b is a factor of a , and a is a multiple of b . For example, 30 = 3 × 10, so 3 and 10 are factors of 30 and 30 is a multiple of 3 and 10

Take note that 1 is a factor of every number.

Understanding factors and multiples is essential for solving many math problems.

Prime Factors

A factor which is a prime number is called a prime factor . For example, the prime factorization of 180 is 2 × 2 × 3 × 3 × 5

You can use repeated division by prime numbers to obtain the prime factors of a given number.

Greatest Common Factor (GCF)

As the name implies, we need to list the factors and find the greatest one that is common to all the numbers. For example, to get the GCF of 24, 60 and 66:

The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24 The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60

The factors of 66 are 1, 2, 3, 6, 11, 22,33 and 66

Look for the greatest factor that is common to all three numbers - thus 6 is the GCF of 24, 60 and 66.

Least Common Multiple (LCM)

As the name implies, we need to list the multiples and to find the least one that is common to all the numbers. For example, to get the LCM of 3, 6 and 9:

The multiples of 3 are 3, 6, 9, 12, 15, 18, 21 … The multiples of 6 are 6, 12, 18, 24, …

The multiples of 9 are 9, 18, 27, …

Look for the least multiple that is common to all three numbers - thus 18 is the LCM of 3, 6 and 9.

Shortcut To Finding LCM Here is a useful shortcut (also called the ladder method) to finding the LCM of a set of numbers. For example, to find the LCM of 3, 6 and 9, we divide them by any factor of the numbers in the following manner:

How to use the Ladder method to find GCF, LCM and simplifying fractions? Step 1: Write the two numbers on one line. Step 2: Draw the L shape. Step 3: Divide out common prime numbers starting with the smallest. LCM makes an L. GCF is down the left side. Simplified fraction is on the bottom. Example: Find the GCF, LCM and simplified fraction for 24 and 36.

LCM & GCF With the Ladder Method Example: Find the LCM and GCF of 24 and 36.

Difference between greatest common factor and least common multiple Example: Find the GCF and LCM of 16 and 24.

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Solve by Factoring Lessons

Several previous lessons explain the techniques used to factor expressions. This lesson focuses on an imporatant application of those techniques – solving equations.

Why solve by factoring?

The most fundamental tools for solving equations are addition, subtraction, multiplication, and division. These methods work well for equations like x + 2 = 10 – 2x   and   2(x – 4) = 0.

But what about equations where the variable carries an exponent, like x 2 + 3x = 8x – 6? This is where factoring comes in. We will use this equation in the first example.

The Solve by Factoring process will require four major steps:

• Move all terms to one side of the equation, usually the left, using addition or subtraction.
• Factor the equation completely.
• Set each factor equal to zero, and solve.
• List each solution from Step 3 as a solution to the original equation.

First Example

The first step is to move all terms to the left using addition and subtraction. First, we will subtract 8x from each side.

Now, we will add 6 to each side.

With all terms on the left side, we proceed to Step 2.

We identify the left as a trinomial, and factor it accordingly:

We now have two factors, (x – 2) and (x – 3).

We now set each factor equal to zero. The result is two subproblems:

Solving the first subproblem, x – 2 = 0, gives x = 2. Solving the second subproblem, x – 3 = 0, gives x = 3.

The final step is to combine the two previous solutions, x = 2 and x = 3, into one solution for the original problem.

Solve by Factoring: Why does it work?

Examine the equation below:

If you let a = 3, then logivally b must equal 0. Similarly, if you let b = 10, then a must equal 0.

Now try letting a be some other non-zero number. You should observe that as long as a does not equal 0, b must be equal to zero.

To state the observation more generally, “If ab = 0, then either a = 0 or b = 0.” This is an important property of zero which we exploit when solving by factoring.

When the example was factored into (x – 2)(x – 3) = 0, this property was applied to determine that either (x – 2) must equal zero, or (x – 3) must equal zero. Therefore, we were able to create two equations and determine two solutions from this observation.

A Second Example

Move all terms to the left side of the equation. We do this by subtracting 45x from each side.

The next step is to factor the left side completely. We first note that the two terms on the left have a greatest common factor of 5x.

Now, (x 2 – 9) can be factored as a difference between two squares.

We are left with three factors: 5x, (x + 3), and (x – 3). As explained in the “Why does it work?” section, at least one of the three factors must be equal to zero.

Create three subproblems by setting each factor equal to zero.

Solving the first equation gives x = 0. Solving the second equation gives x = -3. And solving the third equation gives x= 3.

The final solution is formed from the solutions to the three subproblems.

Third Example

Steps 1 and 2.

All three terms are already on the left side of the equation, so we may begin factoring. First, we factor out a greatest common factor of 3.

Next, we factor a trinomial.

Finally, we factor the binomial (x 2 – 100) as a difference between two squares.

We proceed by setting each of the four factors equal to zero, resulting in four new equations.

The first equation is invalid, and does not yield a solution. The second equation cannot be solved using basic methods. (x 2 + 4 = 0 actually has two imaginary number solutions, but we will save Imaginary Numbers for another lesson!) Equation 3 has a solution of x = -10, and Equation 4 has a solution of x = 10.

We now include all the solutions we found in a single solution to the original problem:

This may be abbreviated as

Solve By Factoring Resources

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Unit 2: Factors, multiples, and patterns

Factors and multiples.

• Understanding factor pairs (Opens a modal)
• Finding factors of a number (Opens a modal)
• Reasoning about factors and multiples (Opens a modal)
• Finding factors and multiples (Opens a modal)
• Identifying multiples (Opens a modal)
• Factors and multiples (Opens a modal)
• Factor pairs Get 5 of 7 questions to level up!
• Identify factors Get 5 of 7 questions to level up!
• Relate factors and multiples Get 5 of 7 questions to level up!
• Identify multiples Get 3 of 4 questions to level up!

Prime and composite numbers

• Prime numbers (Opens a modal)
• Recognizing prime and composite numbers (Opens a modal)
• Prime and composite numbers intro (Opens a modal)
• Prime and composite numbers review (Opens a modal)
• Identify prime numbers Get 5 of 7 questions to level up!
• Identify composite numbers Get 5 of 7 questions to level up!
• Prime and composite numbers Get 3 of 4 questions to level up!

Math patterns

• Factors and multiples: days of the week (Opens a modal)
• Math patterns: table (Opens a modal)
• Math patterns: toothpicks (Opens a modal)
• Patterns with numbers Get 3 of 4 questions to level up!
• Patterns with shapes Get 3 of 4 questions to level up!

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Factors And Multiples

Here we will learn about factors and multiples, including their definitions, listing factors and multiples, and problem solving with factors and multiples.

There are also factors and multiples worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What are factors and multiples?

Factors and multiples are two different types of numbers.

Factors are numbers that will divide into an integer (a whole number) with no remainder . Another name for a factor is a divisor .

Multiples are the result of multiplying a number by an integer.

There are a finite number of factors of a number.

For example, the factors of 18 are 1,2,3,6,9, and 18.

To find all of the factors of any integer, we write out all of the factor pairs in order.

Step-by-step guide: Factors

The highest common factor (HCF), or greatest common factor, is the largest number that is a factor of two or more numbers.

For example, the highest common factor of the numbers 6,8 and 10 is 2.

Step-by-step guide: Highest common factor

Prime numbers have only two factors, themselves and 1.

Any positive integer that is not a prime number is a composite number. Composite numbers have at least 2 factor pairs.

Step-by-step guide: Prime numbers

There are an infinite number of multiples of a number.

For example, the first 5 multiples of 18 are 18,36,54,72, and 90, but we can continue this list indefinitely.

To calculate a multiple of a number n, we have to multiply it by an integer. 

Step-by-step guide: Multiples

The lowest common multiple (LCM), or least common multiple, is the smallest number that is a multiple of two or more numbers.

For example, the lowest common multiple of the numbers 8 and 10 is 40.

Step-by-step guide: Lowest common multiple

We can use factors and multiples to solve problems involving probability, area, substitution, solving quadratics, and equivalent fractions.

How to list factors

In order to list all of the factor pairs of a number n :

State the pair \bf{1 \times n} .

Write the next smallest factor of \bf{n} and calculate its factor pair.

Repeat until the next factor pair is the same as the previous pair.

Write out the list of factors for \bf{n} .

Explain how to list factors

Factors and multiples worksheet

Get your free factors and multiples worksheet of 20+ questions and answers. Includes reasoning and applied questions.

Related lessons on   factors, multiples and primes

Factors and multiples  is part of our series of lessons to support revision on factors, multiples and primes . You may find it helpful to start with the main  factors, multiples and primes  lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

• Factors, multiples and primes  
• HCF and LCM
• Factor trees
• Prime factors

Factors examples

Example 1: listing factors (odd number).

List the factors of 15.

As n=15, you have the first factor pair 1\times{15}.

2 Write the next smallest factor of \bf{n} and calculate its factor pair.

As 15 is an odd number, 15\div{2} is not an integer and so 2 is not a factor of 15.

15\div{3}=5 and so the next factor pair is 3\times{5}.

3 Repeat until the next factor pair is the same as the previous pair.

So far you have,

As 15 is odd, you cannot divide 15 by any even number and get an integer and so 4 is not a factor of 15.

The next factor to try is 5.

As factors are commutative, 3\times{5}=5\times{3} which is the same as the previous factor pair.

We have now found all of the factor pairs,

4 Write out the list of factors for \bf{n} .

Reading down the first column of factors, and up the second column, the factors of 15 are 1, 3, 5, and 15.

Example 2: listing factors (square number)

List the factors of 16.

As n=16, you have the first factor pair 1\times{16}.

As 16 is an even number, 2 is a factor of 16.

16 \div 2=8 and so the next factor pair is 2 \times 8.

So far we have,

\begin{aligned} &1\times{16}\\\\ &2\times{8} \end{aligned}

16 is not a multiple of 3. 

16 is a multiple of 4,

16 \div 4=4 and so the next factor pair is 4 \times 4.

You have reached a repeated factor and so you have found all of the factor pairs for 16,

\begin{aligned} &1\times{16}\\\\ &2\times{8}\\\\ &4\times{4} \end{aligned}

Reading down the first column of factors, and up the second column, the factors of 16 are 1,2,4,8 and 16.

Notice that as you have a repeated factor, you have an odd number of factors in the list above. This can help to determine that 16 is a square number.

Example 3: listing factors (common factors)

The factors of 21 are 1,3,7, and 21. By finding the factors of 6, determine the common factors of 6 and 21.

As you want to list the common factors of 6 and 21, you need to find the factors of each of them, and then highlight common factors (the numbers that appear in both lists).

You already have the factors of 21, so you just need to find the factor pairs for 6.

The first factor pair is 1\times{6}.

As 6 is an even number, you can divide 6 by 2 and get an integer.

6\div{2}=3 and so the next factor pair is 2\times{3}.

The next factor to try is 3 and you have already used this in the previous factor pair (2\times{3}=3\times{2}).

Therefore we have found all of the factor pairs,

\begin{aligned} &1\times{6}\\\\ &2\times{3} \end{aligned}

The factors of 6 are 1,2,3, and 6.

The factors of 21 are 1,3,7, and 21.

The common factors of 6 and 21 are 1 and 3.

Although you are not asked for it here, you can see that the highest common factor of 6 and 21 is 3.

How to calculate multiples

In order to calculate multiples of a number n :

State the first multiple of \bf{n} .

Calculate the second multiple of \bf{n} .

Continue until you have calculated the number of multiples needed.

Write the solution.

Explain how to calculate multiples

Multiples examples

Example 4: listing multiples (two digit number).

List the first 5 multiples of 12.

The first multiple of 12 is 12\times{1}=12.

The second multiple of 12 is 12\times{2}=24.

The first 5 multiples of 12 are 12,24,36,48, and 60.

Example 5: calculate a specific multiple

What is the 13th multiple of 6?

You only need to calculate the 13th multiple of 6 so you can move on to step 2.

It is worth noting here that we are looking at the multiples of 6.

Move on to step 3.

The 13th multiple of 6 is 6\times{13}=78.

The 13th multiple of 6 is 78.

Example 6: common multiples

Given that the first 5 multiples of 12 are 12,24,36,48, and 60, find a common multiple of 8 and 12.

You have the first 5 multiples of 12 and so you need to find the multiples of 8.

The first multiple of 8 is 8\times{1}=8.

You are looking for a number that is also a multiple of 12. \ 8 is not a multiple of 12 so you need to continue. 

The second multiple of 8 is 8\times{2}=16.

16 is not a multiple of 12 so you need to continue.

24 is a multiple of 12. You have found a common multiple of 8 and 12 so you do not need to continue. 

The first common multiple of 8 and 12 is 24.

If you continued to list the multiples of 8 and 12 you would find other common multiples. 24 is the lowest common multiple of 8 and 12.

Common misconceptions

• Factors and multiples

Factors and multiples are easily mixed up. Remember multiples are the multiplication table, whereas factors are the numbers that go into another number without a remainder. 

• Remember \bf{1} and the number itself for factors 

All numbers are a factor of themselves and 1 is a factor of every number. For example, the factors of 6 are 1,2,3,6 and so 6 is a factor of itself.

• Remember the number itself for multiples

All numbers are a multiple of themselves. For example, the multiples of 6 are 6,12,18,24 and so on and so 6 is a multiple of itself.

Practice factors and multiples questions

1. List the factors of 24.

2 and 12, 3 and 8, 4 and 6

24, 48, 72, 96, and 120

1, 2, 3, 4, 6, 8, 12 and 24

The factor pairs of 24 are,

So the factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.

2. List the factors of 49.

1, 7, and 49

The factor pairs of 49 are,

So the factors of 49 are 1, 7, and 49.

3. The factors of 8 are 1,2,4 and 8. By finding the factors of 30, determine the common factors of 8 and 30.

The factor pairs of 30 are,

So the factors of 30 are \colorbox{yellow}{1}, \colorbox{yellow}{2}, \ 3, \ 5, \ 6, \ 10, \ 15, and 30.

The factors of 8 are \colorbox{yellow}{1}, \colorbox{yellow}{2}, \ 4, and 8.

The common factors of 8 and 30 are 1 and 2.

4. List the first 5 multiples of 30.

So, the first 5 multiples of 30 are 30, 60, 90, 120, and 150.

5. What is the 4th multiple of 15?

6. Determine the first 2 common multiples of 3 and 4.

The first 8 multiples of 3 are 3, \ 6, \ 9, \colorbox{yellow}{12}, \ 15, \ 18, \ 21 and \colorbox{yellow}{24}.

The first 8 multiples of 4 are 4, \ 8, \colorbox{yellow}{12}, \ 16, \ 20, \colorbox{yellow}{24}, \ 28 and 32.

The first common factor of 4 and 3 is 12 and the second is 24.

Factors and multiples GCSE questions

1. Here is a list of numbers,

1, \ 2, \ 3, \ 4, \ 6, \ 10, \ 12, \ 16, \ 24, \ 25 .

(a) Write down the multiples of 4.

(b) Which numbers have a factor of 3?

(c) A common multiple of two numbers is 18. The numbers also have a common factor of 3. Write down the two numbers.

(d) Which number has exactly 5 factors?

(a) 4, 12, 16, 24

(b) 3, 6,12, 24

(c) 3 and 6

2. Bus A and Bus B leave the depot at 7 : 40am.

It takes 40 minutes for Bus A to return to the depot.

It takes 30 minutes for Bus B to return to the depot.

What time will both buses be back at the depot?

Multiples of 30 and 40 listed.

120 minutes = 2 hours

3. (a) The length and width of a rectangle are both integers. How many possible rectangles can be drawn with an area of 24cm^{2}?

(b) An isosceles triangle also has side lengths that are integers. How many triangles can be drawn with a perimeter of 8cm?

Factor pairs of 24  are 1 and 24, 2 and 12, 3 and 8, 4 and 6 .

4 rectangles

1 \ (2cm, \ 3cm, \ 3cm only)

Learning checklist

You have now learned how to:

• Use and understand the terms factors and multiples
• Recognise and use factor pairs and commutativity in mental calculations
• Identify factors including all factor pairs of a given number and common factors of two numbers
• Solve problems involving multiplying and dividing including knowledge of factors and multiples

The next lessons are

• Rounding numbers
• Negative numbers

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Factors and Multiples Worksheets

Welcome to our Factors and Multiples Worksheets. Here you will find a wide range of free Math Worksheets which will help your child to learn to use multiples and factors at a 4th Grade/ 5th Grade level.

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Factors and Multiples Worksheet

These sheets have been designed to support your child with their learning of multiples and factors.

The sheets are graded in order of difficulty with the easiest sheet coming first in each section.

Using these sheets will help your child to:

• Know and understand what multiples and factors are;
• apply knowledge of multiples and factors to solve problems;
• Develop and practice their mental calculation skills.

Want to test yourself to see how well you have understood this skill?.

• Try our NEW quick quiz at the bottom of this page.

Quicklinks to ...

Multiples and Factors Help

Multiples worksheets, factors worksheets.

• Factors and Multiples Riddles
• Easier/Harder Worksheets
• More related resources

Factors and Multiples Online Quiz

If a number is a multiple of another number, it means that it can be made out of adding groups of the other number together.

12 is a multiple of 4 because 4 + 4 + 4 = 12 (or 4 x 3 = 12)

27 is a multiple of 9 because 9 + 9 + 9 = 27 (or 9 x 3 = 27)

17 is not a multiple of 4 because it cannot be made by adding groups of 4 together.

A factor is a number that divides into another number with no remainder.

In other words every number is divisible by each of its factors.

1 is a factor of every whole number, because every integer is divisible by one.

3 and 7 are both factors of 21 because 3 x 7 = 21

10 and 6 are both factors of 60 because 10 x 6 = 60

7 is not a factor of 24 because 24 is not divisible by 7 (24 ÷ 7 = 3 remainder 3).

Multiples and Factors are connected with each other:

• if we know that 3 is a factor of 12, then 12 is a multiple of 3
• if we know that 33 is a multiple of 11, then 11 is a factor of 33.
• also, if we know that 24 is not a multiple of 7, then 7 is not factor of 24.

The example below shows the relationship visually.

If we know that 3 is a factor of 24, then 24 must be a multiple of 3.

If we know that 24 is a multiple of 3, then 3 must be a factor of 24.

Multiples and Factors Worksheets

We have split our worksheets into 3 different sections:

• the first section contains only worksheets about Multiples
• the second section contains only worksheets about Factors
• the third section contains worksheets with both Factors and Multiples
• Multiples Sheet 4:1
• PDF version
• Multiples Sheet 4:2
• Multiples Sheet 4:3
• Multiples Sheet 4:4

We have two worksheets on finding Factor Pairs up to 100.

We have two worksheets which involve finding all the factors of different numbers.

• Factor Pairs Worksheet 1
• Factor Pairs Worksheet 2
• Factors Worksheet 4:1
• Factors Worksheet 4:2
• Factors and Multiples Worksheet 4:1
• Factors and Multiples Worksheet 4:2
• Factors and Multiples Worksheet 4:3

Factor and Multiples Riddles

Using riddles is a great way to get children to apply their knowledge of factors and multiples to solve problems.

It is also a good way to get children working collaboratively and talking about the language together.

Each riddle consists of some clues and a selection of possible answers.

Solving the clues gradually eliminates all the incorrect answers leaving just one solution.

• Factors and Multiples Riddles 1
• Factors and Multiples Riddles 2

Looking for some easier Multiples Sheets

The sheets in this section cover similar areas to the worksheets on this page but are at an easier level.

• round a number to the nearest 10, 100 or 1000;
• use the > and < symbols correctly for inequalities;
• use multiples and apply them to solve problems.
• Rounding Inequalities Multiples Worksheets

Looking for some harder Factors and Multiples Worksheets

We also have some more advanced worksheets about multiples and factors.

The worksheets below are more suitable for 6th graders and above.

• Greatest Common Factor Worksheets
• Least Common Multiple Worksheets
• Factor Tree Worksheets (easier)
• Prime Factorization Worksheets (harder)

More Recommended Math Resources

Take a look at some more of our worksheets similar to these.

Divisibility Rules 1-10 Chart

We have a range of charts which can help you determine whether a number between 1 and 10 is a factor of a number.

• Divisibility Rules 1-10 Charts

Balancing Math Equations Worksheets

The sheets in this area will help your child understand the use and purpose of the equals sign (=) in an equation.

It will also help children learn to start manipulating and calculating numerical expressions so that they are equivalent.

This will stand them in good stead for when they start to learn algebra, and manipulate algebraic equations.

• Balancing Math Equations

Sieve of Erastosthenes

The Sieve of Erastosthenes is a method for finding what is a prime numbers between 2 and any given number.

Eratosthenes was a Greek mathematician (as well as being a poet, an astronomer and musician) who lived from about 276BC to 194BC.

If you want to find out more about his sieve for finding primes, and print out some Sieve of Eratosthenes worksheets, use the link below.

• Sieve of Eratosthenes page

Want to find out more about primes?

Take a look at our Prime Number page which clearly describes what a prime numbers is and what they are not.

There are also many different questions about prime numbers answered, as well as information about the density of primes.

• What is a Prime Number

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This quick quiz tests your knowledge and skill with factors and multiples here!

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• Solve equations and inequalities
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The process of factoring is essential to the simplification of many algebraic expressions and is a useful tool in solving higher degree equations. In fact, the process of factoring is so important that very little of algebra beyond this point can be accomplished without understanding it.

In earlier chapters the distinction between terms and factors has been stressed. You should remember that terms are added or subtracted and factors are multiplied. Three important definitions follow.

Terms occur in an indicated sum or difference. Factors occur in an indicated product.

• An expression is in factored form only if the entire expression is an indicated product.

Note in these examples that we must always regard the entire expression. Factors can be made up of terms and terms can contain factors, but factored form must conform to the definition above.

Factoring is a process of changing an expression from a sum or difference of terms to a product of factors.

Note that in this definition it is implied that the value of the expression is not changed - only its form.

REMOVING COMMON FACTORS

• Determine which factors are common to all terms in an expression.
• Factor common factors.

In the previous chapter we multiplied an expression such as 5(2x + 1) to obtain 10x + 5. In general, factoring will "undo" multiplication. Each term of 10x + 5 has 5 as a factor, and 10x + 5 = 5(2x + 1).

To factor an expression by removing common factors proceed as in example 1.

Next look for factors that are common to all terms, and search out the greatest of these. This is the greatest common factor. In this case, the greatest common factor is 3x.

Proceed by placing 3x before a set of parentheses.

The terms within the parentheses are found by dividing each term of the original expression by 3x.

If we had only removed the factor "3" from 3x 2 + 6xy + 9xy 2 , the answer would be

3(x 2 + 2xy + 3xy 2 ).

Multiplying to check, we find the answer is actually equal to the original expression. However, the factor x is still present in all terms. Hence, the expression is not completely factored.

• It must be possible to multiply the factored expression and get the original expression.
• FThe expression must be completely factored.

Example 2 Factor 12x 3 + 6x 2 + 18x.

At this point it should not be necessary to list the factors of each term. You should be able to mentally determine the greatest common factor. A good procedure to follow is to think of the elements individually. In other words, don�t attempt to obtain all common factors at once but get first the number, then each letter involved. For instance, 6 is a factor of 12, 6, and 18, and x is a factor of each term. Hence 12x 3 + 6x 2 + 18x = 6x(2x 2 + x + 3). Multiplying, we get the original and can see that the terms within the parentheses have no other common factor, so we know the solution is correct.

If an expression cannot be factored it is said to be prime .

FACTORING BY GROUPING

• Factor expressions when the common factor involves more than one term.
• Factor by grouping.

An extension of the ideas presented in the previous section applies to a method of factoring called grouping .

First we must note that a common factor does not need to be a single term. For instance, in the expression 2y(x + 3) + 5(x + 3) we have two terms. They are 2y(x + 3) and 5(x + 3). In each of these terms we have a factor (x + 3) that is made up of terms. This factor (x + 3) is a common factor.

Sometimes when there are four or more terms, we must insert an intermediate step or two in order to factor.

First note that not all four terms in the expression have a common factor, but that some of them do. For instance, we can factor 3 from the first two terms, giving 3(ax + 2y). If we factor a from the remaining two terms, we get a(ax + 2y). The expression is now 3(ax + 2y) + a(ax + 2y), and we have a common factor of (ax + 2y) and can factor as (ax + 2y)(3 + a). Multiplying (ax + 2y)(3 + a), we get the original expression 3ax + 6y + a 2 x + 2ay and see that the factoring is correct.

This is an example of factoring by grouping since we "grouped" the terms two at a time.

Sometimes the terms must first be rearranged before factoring by grouping can be accomplished.

Example 7 Factor 3ax + 2y + 3ay + 2x.

The first two terms have no common factor, but the first and third terms do, so we will rearrange the terms to place the third term after the first. Always look ahead to see the order in which the terms could be arranged.

In all cases it is important to be sure that the factors within parentheses are exactly alike. This may require factoring a negative number or letter.

Example 8 Factor ax - ay - 2x + 2y.

Note that when we factor a from the first two terms, we get a(x - y). Looking at the last two terms, we see that factoring +2 would give 2(-x + y) but factoring "-2" gives - 2(x - y). We want the terms within parentheses to be (x - y), so we proceed in this manner.

FACTORING TRINOMIALS

• Mentally multiply two binomials.
• Factor a trinomial having a first term coefficient of 1.
• Find the factors of any factorable trinomial.

A large number of future problems will involve factoring trinomials as products of two binomials. In the previous chapter you learned how to multiply polynomials. We now wish to look at the special case of multiplying two binomials and develop a pattern for this type of multiplication.

Since this type of multiplication is so common, it is helpful to be able to find the answer without going through so many steps. Let us look at a pattern for this.

From the example (2x + 3)(3x - 4) = 6x 2 + x - 12, note that the first term of the answer (6x 2 ) came from the product of the two first terms of the factors, that is (2x)(3x).

Also note that the third term (-12) came from the product of the second terms of the factors, that is ( + 3)(-4).

We now have the following part of the pattern:

Now looking at the example again, we see that the middle term (+x) came from a sum of two products (2x)( -4) and (3)(3x).

• First term by first term
• Outside terms
• Inside terms
• Last term by last term

These products are shown by this pattern.

When the products of the outside terms and inside terms give like terms, they can be combined and the solution is a trinomial.

You should memorize this pattern.

Not only should this pattern be memorized, but the student should also learn to go from problem to answer without any written steps. This mental process of multiplying is necessary if proficiency in factoring is to be attained.

As you work the following exercises, attempt to arrive at a correct answer without writing anything except the answer. The more you practice this process, the better you will be at factoring.

Now that we have established the pattern of multiplying two binomials, we are ready to factor trinomials. We will first look at factoring only those trinomials with a first term coefficient of 1.

Since this is a trinomial and has no common factor we will use the multiplication pattern to factor.

First write parentheses under the problem.

We now wish to fill in the terms so that the pattern will give the original trinomial when we multiply. The first term is easy since we know that (x)(x) = x 2 .

We must now find numbers that multiply to give 24 and at the same time add to give the middle term. Notice that in each of the following we will have the correct first and last term.

Only the last product has a middle term of 11x, and the correct solution is

This method of factoring is called trial and error - for obvious reasons.

Here the problem is only slightly different. We must find numbers that multiply to give 24 and at the same time add to give - 11. You should always keep the pattern in mind. The last term is obtained strictly by multiplying, but the middle term comes finally from a sum. Knowing that the product of two negative numbers is positive, but the sum of two negative numbers is negative, we obtain

We are here faced with a negative number for the third term, and this makes the task slightly more difficult. Since -24 can only be the product of a positive number and a negative number, and since the middle term must come from the sum of these numbers, we must think in terms of a difference. We must find numbers whose product is 24 and that differ by 5. Furthermore, the larger number must be negative, because when we add a positive and negative number the answer will have the sign of the larger. Keeping all of this in mind, we obtain

• When the sign of the third term is positive, both signs in the factors must be alike-and they must be like the sign of the middle term.
• When the sign of the last term is negative, the signs in the factors must be unlike-and the sign of the larger must be like the sign of the middle term.

In the previous exercise the coefficient of each of the first terms was 1. When the coefficient of the first term is not 1, the problem of factoring is much more complicated because the number of possibilities is greatly increased.

Notice that there are twelve ways to obtain the first and last terms, but only one has 17x as a middle term.

There is only one way to obtain all three terms:

In this example one out of twelve possibilities is correct. Thus trial and error can be very time-consuming.

Even though the method used is one of guessing, it should be "educated guessing" in which we apply all of our knowledge about numbers and exercise a great deal of mental arithmetic. In the preceding example we would immediately dismiss many of the combinations. Since we are searching for 17x as a middle term, we would not attempt those possibilities that multiply 6 by 6, or 3 by 12, or 6 by 12, and so on, as those products will be larger than 17. Also, since 17 is odd, we know it is the sum of an even number and an odd number. All of these things help reduce the number of possibilities to try.

• The last term is positive, so two like signs.
• The middle term is negative, so both signs will be negative.
• The factors of 6x2 are x, 2x, 3x, 6x. The factors of 15 are 1, 3, 5, 15.
• Eliminate as too large the product of 15 with 2x, 3x, or 6x. Try some reasonable combinations.

• The last term is negative, so unlike signs.
• We must find products that differ by 5 with the larger number negative.
• We eliminate a product of 4x and 6 as probably too large.
• Try some combinations.

(4x - 3)(x + 2) : Here the middle term is + 5x, which is the right number but the wrong sign. Be careful not to accept this as the solution, but switch signs so the larger product agrees in sign with the middle term.

SPECIAL CASES IN FACTORING

• Identify and factor the differences of two perfect squares.
• Identify and factor a perfect square trinomial.

In this section we wish to examine some special cases of factoring that occur often in problems. If these special cases are recognized, the factoring is then greatly simplified.

The first special case we will discuss is the difference of two perfect squares .

Recall that in multiplying two binomials by the pattern, the middle term comes from the sum of two products.

From our experience with numbers we know that the sum of two numbers is zero only if the two numbers are negatives of each other.

In each example the middle term is zero. Note that if two binomials multiply to give a binomial (middle term missing), they must be in the form of (a - b) (a + b).

Here both terms are perfect squares and they are separated by a negative sign.

Special cases do make factoring easier, but be certain to recognize that a special case is just that-very special. In this case both terms must be perfect squares and the sign must be negative, hence "the difference of two perfect squares."

You must also be careful to recognize perfect squares. Remember that perfect square numbers are numbers that have square roots that are integers. Also, perfect square exponents are even.

Another special case in factoring is the perfect square trinomial. Observe that squaring a binomial gives rise to this case.

• The first term is a perfect square.
• The third term is a perfect square.
• The middle term is twice the product of the square root of the first and third terms.
• 25x 2 is a perfect square-principal square root = 5x.
• 4 is a perfect square-principal square root = 2.
• 20x is twice the product of the square roots of 25x 2 and
• 20x = 2(5x)(2).

To factor a perfect square trinomial form a binomial with the square root of the first term, the square root of the last term, and the sign of the middle term, and indicate the square of this binomial.

Thus, 25x 2 + 20x + 4 = (5x + 2) 2

Not the special case of a perfect square trinomial.

OPTIONAL SHORTCUTS TO TRIAL AND ERROR FACTORING

• Find the key number of a trinomial.
• Use the key number to factor a trinomial.

In this section we wish to discuss some shortcuts to trial and error factoring. These are optional for two reasons. First, some might prefer to skip these techniques and simply use the trial and error method; second, these shortcuts are not always practical for large numbers. However, they will increase speed and accuracy for those who master them.

The first step in these shortcuts is finding the key number . After you have found the key number it can be used in more than one way.

In a trinomial to be factored the key number is the product of the coefficients of the first and third terms.

The first use of the key number is shown in example 3.

A second use for the key number as a shortcut involves factoring by grouping. It works as in example 5.

COMPLETE FACTORIZATION

• First look for common factors.
• Factor the remaining trinomial by applying the methods of this chapter.

We have now studied all of the usual methods of factoring found in elementary algebra. However, you must be aware that a single problem can require more than one of these methods. Remember that there are two checks for correct factoring.

• Will the factors multiply to give the original problem?
• Are all factors prime?

A good procedure to follow in factoring is to always remove the greatest common factor first and then factor what remains, if possible.

• Factoring is a process that changes a sum or difference of terms to a product of factors.
• A prime expression cannot be factored.
• The greatest common factor is the greatest factor common to all terms.
• An expression is completely factored when no further factoring is possible.
• The possibility of factoring by grouping exists when an expression contains four or more terms.
• The FOIL method can be used to multiply two binomials.
• Special cases in factoring include the difference of two squares and perfect square trinomials .
• The key number is the product of the coefficients of the first and third terms of a trinomial.
• To remove common factors find the greatest common factor and divide each term by it.

• To factor a perfect square trinomial form a binomial with the square root of the first term, the square root of the last term, and the sign of the middle term and indicate the square of this binomial.
• Use the key number as an aid in determining factors whose sum is the coefficient of the middle term of a trinomial.

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What is Problem Solving? (Steps, Techniques, Examples)

By Status.net Editorial Team on May 7, 2023 — 5 minutes to read

What Is Problem Solving?

Definition and importance.

Problem solving is the process of finding solutions to obstacles or challenges you encounter in your life or work. It is a crucial skill that allows you to tackle complex situations, adapt to changes, and overcome difficulties with ease. Mastering this ability will contribute to both your personal and professional growth, leading to more successful outcomes and better decision-making.

Problem-Solving Steps

The problem-solving process typically includes the following steps:

• Identify the issue : Recognize the problem that needs to be solved.
• Analyze the situation : Examine the issue in depth, gather all relevant information, and consider any limitations or constraints that may be present.
• Generate potential solutions : Brainstorm a list of possible solutions to the issue, without immediately judging or evaluating them.
• Evaluate options : Weigh the pros and cons of each potential solution, considering factors such as feasibility, effectiveness, and potential risks.
• Select the best solution : Choose the option that best addresses the problem and aligns with your objectives.
• Implement the solution : Put the selected solution into action and monitor the results to ensure it resolves the issue.
• Review and learn : Reflect on the problem-solving process, identify any improvements or adjustments that can be made, and apply these learnings to future situations.

Defining the Problem

To start tackling a problem, first, identify and understand it. Analyzing the issue thoroughly helps to clarify its scope and nature. Ask questions to gather information and consider the problem from various angles. Some strategies to define the problem include:

• Brainstorming with others
• Asking the 5 Ws and 1 H (Who, What, When, Where, Why, and How)
• Analyzing cause and effect
• Creating a problem statement

Generating Solutions

Once the problem is clearly understood, brainstorm possible solutions. Think creatively and keep an open mind, as well as considering lessons from past experiences. Consider:

• Creating a list of potential ideas to solve the problem
• Grouping and categorizing similar solutions
• Prioritizing potential solutions based on feasibility, cost, and resources required
• Involving others to share diverse opinions and inputs

Evaluating and Selecting Solutions

Evaluate each potential solution, weighing its pros and cons. To facilitate decision-making, use techniques such as:

• SWOT analysis (Strengths, Weaknesses, Opportunities, Threats)
• Decision-making matrices
• Pros and cons lists
• Risk assessments

After evaluating, choose the most suitable solution based on effectiveness, cost, and time constraints.

Implementing and Monitoring the Solution

Implement the chosen solution and monitor its progress. Key actions include:

• Communicating the solution to relevant parties
• Setting timelines and milestones
• Assigning tasks and responsibilities
• Monitoring the solution and making adjustments as necessary
• Evaluating the effectiveness of the solution after implementation

Utilize feedback from stakeholders and consider potential improvements. Remember that problem-solving is an ongoing process that can always be refined and enhanced.

Problem-Solving Techniques

During each step, you may find it helpful to utilize various problem-solving techniques, such as:

• Brainstorming : A free-flowing, open-minded session where ideas are generated and listed without judgment, to encourage creativity and innovative thinking.
• Root cause analysis : A method that explores the underlying causes of a problem to find the most effective solution rather than addressing superficial symptoms.
• SWOT analysis : A tool used to evaluate the strengths, weaknesses, opportunities, and threats related to a problem or decision, providing a comprehensive view of the situation.
• Mind mapping : A visual technique that uses diagrams to organize and connect ideas, helping to identify patterns, relationships, and possible solutions.

Brainstorming

When facing a problem, start by conducting a brainstorming session. Gather your team and encourage an open discussion where everyone contributes ideas, no matter how outlandish they may seem. This helps you:

• Generate a diverse range of solutions
• Encourage all team members to participate
• Foster creative thinking

When brainstorming, remember to:

• Reserve judgment until the session is over
• Encourage wild ideas
• Combine and improve upon ideas

Root Cause Analysis

For effective problem-solving, identifying the root cause of the issue at hand is crucial. Try these methods:

• 5 Whys : Ask “why” five times to get to the underlying cause.
• Fishbone Diagram : Create a diagram representing the problem and break it down into categories of potential causes.
• Pareto Analysis : Determine the few most significant causes underlying the majority of problems.

SWOT Analysis

SWOT analysis helps you examine the Strengths, Weaknesses, Opportunities, and Threats related to your problem. To perform a SWOT analysis:

• List your problem’s strengths, such as relevant resources or strong partnerships.
• Identify its weaknesses, such as knowledge gaps or limited resources.
• Explore opportunities, like trends or new technologies, that could help solve the problem.
• Recognize potential threats, like competition or regulatory barriers.

SWOT analysis aids in understanding the internal and external factors affecting the problem, which can help guide your solution.

Mind Mapping

A mind map is a visual representation of your problem and potential solutions. It enables you to organize information in a structured and intuitive manner. To create a mind map:

• Write the problem in the center of a blank page.
• Draw branches from the central problem to related sub-problems or contributing factors.
• Add more branches to represent potential solutions or further ideas.

Mind mapping allows you to visually see connections between ideas and promotes creativity in problem-solving.

Examples of Problem Solving in Various Contexts

In the business world, you might encounter problems related to finances, operations, or communication. Applying problem-solving skills in these situations could look like:

• Identifying areas of improvement in your company’s financial performance and implementing cost-saving measures
• Resolving internal conflicts among team members by listening and understanding different perspectives, then proposing and negotiating solutions
• Streamlining a process for better productivity by removing redundancies, automating tasks, or re-allocating resources

In educational contexts, problem-solving can be seen in various aspects, such as:

• Addressing a gap in students’ understanding by employing diverse teaching methods to cater to different learning styles
• Developing a strategy for successful time management to balance academic responsibilities and extracurricular activities
• Seeking resources and support to provide equal opportunities for learners with special needs or disabilities

Everyday life is full of challenges that require problem-solving skills. Some examples include:

• Overcoming a personal obstacle, such as improving your fitness level, by establishing achievable goals, measuring progress, and adjusting your approach accordingly
• Navigating a new environment or city by researching your surroundings, asking for directions, or using technology like GPS to guide you
• Dealing with a sudden change, like a change in your work schedule, by assessing the situation, identifying potential impacts, and adapting your plans to accommodate the change.
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Problem-Solving Strategies and Obstacles

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• Application
• Improvement

From deciding what to eat for dinner to considering whether it's the right time to buy a house, problem-solving is a large part of our daily lives. Learn some of the problem-solving strategies that exist and how to use them in real life, along with ways to overcome obstacles that are making it harder to resolve the issues you face.

What Is Problem-Solving?

In cognitive psychology , the term 'problem-solving' refers to the mental process that people go through to discover, analyze, and solve problems.

A problem exists when there is a goal that we want to achieve but the process by which we will achieve it is not obvious to us. Put another way, there is something that we want to occur in our life, yet we are not immediately certain how to make it happen.

Maybe you want a better relationship with your spouse or another family member but you're not sure how to improve it. Or you want to start a business but are unsure what steps to take. Problem-solving helps you figure out how to achieve these desires.

The problem-solving process involves:

• Discovery of the problem
• Deciding to tackle the issue
• Seeking to understand the problem more fully
• Researching available options or solutions
• Taking action to resolve the issue

Before problem-solving can occur, it is important to first understand the exact nature of the problem itself. If your understanding of the issue is faulty, your attempts to resolve it will also be incorrect or flawed.

Problem-Solving Mental Processes

Several mental processes are at work during problem-solving. Among them are:

• Perceptually recognizing the problem
• Representing the problem in memory
• Considering relevant information that applies to the problem
• Identifying different aspects of the problem
• Labeling and describing the problem

Problem-Solving Strategies

There are many ways to go about solving a problem. Some of these strategies might be used on their own, or you may decide to employ multiple approaches when working to figure out and fix a problem.

An algorithm is a step-by-step procedure that, by following certain "rules" produces a solution. Algorithms are commonly used in mathematics to solve division or multiplication problems. But they can be used in other fields as well.

In psychology, algorithms can be used to help identify individuals with a greater risk of mental health issues. For instance, research suggests that certain algorithms might help us recognize children with an elevated risk of suicide or self-harm.

One benefit of algorithms is that they guarantee an accurate answer. However, they aren't always the best approach to problem-solving, in part because detecting patterns can be incredibly time-consuming.

There are also concerns when machine learning is involved—also known as artificial intelligence (AI)—such as whether they can accurately predict human behaviors.

Heuristics are shortcut strategies that people can use to solve a problem at hand. These "rule of thumb" approaches allow you to simplify complex problems, reducing the total number of possible solutions to a more manageable set.

If you find yourself sitting in a traffic jam, for example, you may quickly consider other routes, taking one to get moving once again. When shopping for a new car, you might think back to a prior experience when negotiating got you a lower price, then employ the same tactics.

While heuristics may be helpful when facing smaller issues, major decisions shouldn't necessarily be made using a shortcut approach. Heuristics also don't guarantee an effective solution, such as when trying to drive around a traffic jam only to find yourself on an equally crowded route.

Trial and Error

A trial-and-error approach to problem-solving involves trying a number of potential solutions to a particular issue, then ruling out those that do not work. If you're not sure whether to buy a shirt in blue or green, for instance, you may try on each before deciding which one to purchase.

This can be a good strategy to use if you have a limited number of solutions available. But if there are many different choices available, narrowing down the possible options using another problem-solving technique can be helpful before attempting trial and error.

In some cases, the solution to a problem can appear as a sudden insight. You are facing an issue in a relationship or your career when, out of nowhere, the solution appears in your mind and you know exactly what to do.

Insight can occur when the problem in front of you is similar to an issue that you've dealt with in the past. Although, you may not recognize what is occurring since the underlying mental processes that lead to insight often happen outside of conscious awareness .

Research indicates that insight is most likely to occur during times when you are alone—such as when going on a walk by yourself, when you're in the shower, or when lying in bed after waking up.

How to Apply Problem-Solving Strategies in Real Life

If you're facing a problem, you can implement one or more of these strategies to find a potential solution. Here's how to use them in real life:

• Create a flow chart . If you have time, you can take advantage of the algorithm approach to problem-solving by sitting down and making a flow chart of each potential solution, its consequences, and what happens next.
• Recall your past experiences . When a problem needs to be solved fairly quickly, heuristics may be a better approach. Think back to when you faced a similar issue, then use your knowledge and experience to choose the best option possible.
• Start trying potential solutions . If your options are limited, start trying them one by one to see which solution is best for achieving your desired goal. If a particular solution doesn't work, move on to the next.
• Take some time alone . Since insight is often achieved when you're alone, carve out time to be by yourself for a while. The answer to your problem may come to you, seemingly out of the blue, if you spend some time away from others.

Obstacles to Problem-Solving

Problem-solving is not a flawless process as there are a number of obstacles that can interfere with our ability to solve a problem quickly and efficiently. These obstacles include:

• Assumptions: When dealing with a problem, people can make assumptions about the constraints and obstacles that prevent certain solutions. Thus, they may not even try some potential options.
• Functional fixedness : This term refers to the tendency to view problems only in their customary manner. Functional fixedness prevents people from fully seeing all of the different options that might be available to find a solution.
• Irrelevant or misleading information: When trying to solve a problem, it's important to distinguish between information that is relevant to the issue and irrelevant data that can lead to faulty solutions. The more complex the problem, the easier it is to focus on misleading or irrelevant information.
• Mental set: A mental set is a tendency to only use solutions that have worked in the past rather than looking for alternative ideas. A mental set can work as a heuristic, making it a useful problem-solving tool. However, mental sets can also lead to inflexibility, making it more difficult to find effective solutions.

How to Improve Your Problem-Solving Skills

In the end, if your goal is to become a better problem-solver, it's helpful to remember that this is a process. Thus, if you want to improve your problem-solving skills, following these steps can help lead you to your solution:

• Recognize that a problem exists . If you are facing a problem, there are generally signs. For instance, if you have a mental illness , you may experience excessive fear or sadness, mood changes, and changes in sleeping or eating habits. Recognizing these signs can help you realize that an issue exists.
• Decide to solve the problem . Make a conscious decision to solve the issue at hand. Commit to yourself that you will go through the steps necessary to find a solution.
• Seek to fully understand the issue . Analyze the problem you face, looking at it from all sides. If your problem is relationship-related, for instance, ask yourself how the other person may be interpreting the issue. You might also consider how your actions might be contributing to the situation.
• Research potential options . Using the problem-solving strategies mentioned, research potential solutions. Make a list of options, then consider each one individually. What are some pros and cons of taking the available routes? What would you need to do to make them happen?
• Take action . Select the best solution possible and take action. Action is one of the steps required for change . So, go through the motions needed to resolve the issue.
• Try another option, if needed . If the solution you chose didn't work, don't give up. Either go through the problem-solving process again or simply try another option.

You can find a way to solve your problems as long as you keep working toward this goal—even if the best solution is simply to let go because no other good solution exists.

Sarathy V. Real world problem-solving .  Front Hum Neurosci . 2018;12:261. doi:10.3389/fnhum.2018.00261

Dunbar K. Problem solving . A Companion to Cognitive Science . 2017. doi:10.1002/9781405164535.ch20

Stewart SL, Celebre A, Hirdes JP, Poss JW. Risk of suicide and self-harm in kids: The development of an algorithm to identify high-risk individuals within the children's mental health system . Child Psychiat Human Develop . 2020;51:913-924. doi:10.1007/s10578-020-00968-9

Rosenbusch H, Soldner F, Evans AM, Zeelenberg M. Supervised machine learning methods in psychology: A practical introduction with annotated R code . Soc Personal Psychol Compass . 2021;15(2):e12579. doi:10.1111/spc3.12579

Mishra S. Decision-making under risk: Integrating perspectives from biology, economics, and psychology . Personal Soc Psychol Rev . 2014;18(3):280-307. doi:10.1177/1088868314530517

Csikszentmihalyi M, Sawyer K. Creative insight: The social dimension of a solitary moment . In: The Systems Model of Creativity . 2015:73-98. doi:10.1007/978-94-017-9085-7_7

Chrysikou EG, Motyka K, Nigro C, Yang SI, Thompson-Schill SL. Functional fixedness in creative thinking tasks depends on stimulus modality .  Psychol Aesthet Creat Arts . 2016;10(4):425‐435. doi:10.1037/aca0000050

Huang F, Tang S, Hu Z. Unconditional perseveration of the short-term mental set in chunk decomposition .  Front Psychol . 2018;9:2568. doi:10.3389/fpsyg.2018.02568

National Alliance on Mental Illness. Warning signs and symptoms .

Mayer RE. Thinking, problem solving, cognition, 2nd ed .

Schooler JW, Ohlsson S, Brooks K. Thoughts beyond words: When language overshadows insight. J Experiment Psychol: General . 1993;122:166-183. doi:10.1037/0096-3445.2.166

By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

12 Apr What Affects Problem Solving

Many factors affect the problem solving process and hence it can become complicated and drawn out when they are unaccounted for. Acknowledging the factors that affect the process and taking them into account when forming a solution gives teams the best chance of solving the problem effectively. Below we have outlined the key factors affecting the problem solving process.

Understanding the problem 

The most important factor in solving a problem is to first fully understand it. This includes understanding the bigger picture it sits within, the factors and stakeholders involved, the causes of the problem and any potential solutions. Effective solutions are unlikely to be discovered if the exact problem is not fully understood.

Personality types/Temperament 

McCauley (1987) was one of the first authors to link personalities to problem solving skills. Attributes like patience, communication, team skills and cognitive skills can all affect an individual’s likelihood of solving a problem. Different individuals will take different approaches to solving problems and experience varying degrees of success. For this reason, as a manager, it is important to select team members for a project whose skills align with the problem at hand.

Skills/Competencies

Individual’s skills will also affect the problem solving process. For example, a straight-forward technical issue may appear very complicated to an individual from a non-technical background. Skill levels are most commonly determined by experience and training and for this reason it is important to expose newer team members to a wide variety of problems, as well as providing training.

Resources available

Although many individuals believe they have the capabilities to solve a certain problem, the resources available to them can often slow-down the process. These resources may be in the form of technology, human capital or finance. For example, a team may come up with a solution for an inefficient transport system by suggesting new vehicles are purchased. Despite the solution solving the problem entirely, it may not fit within the budget. This is why only realistic solutions should be pursued and resources should not be wasted on other projects.

External factors should also always be taken into account when solving a problem, as factors that may not seem to directly affect the problem can often play a part. Examples include competitor actions, fluctuations in the economy, government restrictions and environmental issues.

Carskadon, Thomas G, Nancy G McCarley, and Mary H McCaulley. (1987). Compendium of Research Involving the Myers-Briggs Type Indicator . Gainesville, Fl.: Center for Applications of Psychological Type, 1987. Print.

• Math Article

Common Factors

A factor of a number is an exact divisor of the given number. Every factor of a number is less than or equal to the given number, i.e. it cannot be greater than the given number. Every number has at least two factors, some numbers have more than two factors. For example, 1, 2, 3, and 6 are the factors of 6. Also, 1 is a factor of every number and every number is a factor in itself. It can be said that the number of factors of a given number is finite. Also, check the highest common factor  for any number here.

Common Factors Definition

In Maths, common factors are defined as factors that are common to two or more numbers. In other words, a common factor is a number with which a set of two or more numbers will be divided exactly.

Common Factors of Two Numbers

To find common factors of two numbers, first, list out all the factors of two numbers separately and then compare them. Now write the factors which are common to both the numbers. These factors are called common factors of given two numbers.

How to Find Common Factors

As we know, the factors are the numbers that divide the original number completely. But how to check if two or more numbers have common factors between them.

Follow the below steps to find the common factors.

• Write the factors of the given numbers.
• Find the common factor present in them.

Let us see some examples here.

Common factors of 15 and 25

Let us check the factors of the two numbers, i.e., 15 and 25.

15 = 1, 3, 5, 15

25 = 1, 5, 25

We can see that both 15 and 25 have 5 as the common factor.

Common Factors of 12 and 18

First, we need to write all the factors of 12 and 18.

Factors of 12 = 1,2,3,4,6, 12

Factors of 18 = 1,2,3,6,9, 18

Clearly, we can see that the common factors between 12 and 18 are 1,2,3 and 6.

Common Factors of 8 and 24

Let us find the factors of 8 and 24.

Factors of 8 = 1,2,4,8

Factors of 24 = 1,2,3,4,6,8,12,24

So we can see here that the common factors of 8 and 24 are 1,2,4 and 8.

• Factors And Multiples
• Factors Of A Number
• Number System For Class 9

Common Factors Examples

Understand more about common factors with the below examples.

Example 1: Find the common factors of 36 and 63.

1 × 36 = 36

2 × 18 = 36

3 × 12 = 36

Stop here, since the number 6 is repeated.

1, 2, 3, 4, 6, 9, 12, 18, and 36 are factors of 36.

1 × 63 = 63

3 × 21 = 63

Stop here, since the numbers 7 and 9 are repeated.

1, 3, 7, 9, 21 and 63 are factors of 63.

1, 3 and 9 are common in both the lists.

Hence, the common factors of 36 and 63 are 1, 3, 9.

We can also find the common factors for more than two numbers. Consider the below example to understand the process of finding common factors of three numbers.

Example 2: What are the common factors of 45, 80 and 28?

1 × 45 = 45

3 × 15 = 45

Stop here, since the numbers 5 and 9 are repeated.

1, 3, 5, 9, 15 and 45 are factors of 45.

1 × 80 = 80

2 × 40 = 80

4 × 20 = 80

5 × 16 = 80

8 × 10 = 80

10 × 8 = 80

Stop here, since the numbers 8 and 10 are repeated.

1, 2, 4, 5, 8, 10, 16, 20, 40 and 80 are factors of 80.

1 × 28 = 28

2 × 14 = 28

Stop here, since the numbers 4 and 7 are repeated.

1, 2, 4, 7, 14 and 28 are factors of 28.

Only 1 is common in the above lists.

Hence, the common factor of 45, 80 and 28 is 1.

Applications

Common factors are used in solving problems like How to simplify fractions ?

Consider the fraction 40/96.

40/96 = 5/ 12

Since, the common factors of 40 and 96 are 1, 2, 4, 8.

Select the greatest among them and express the numbers of the fraction as a multiple of this greatest number.

96 = 8 × 12

Other applications like comparing prices, understanding time-distance concepts and time & work problems.

Practice Questions

Go through the questions given below and try to solve the problems for a better understanding of the concept.

• What are the common factors of 18 and 30?
• What are the common factors of 15 and 25?
• What are the common factors of 18 and 27?
• What are the common factors of 18 and 21?
• What are the common factors of 18 and 24?

Frequently Asked Questions on Common Factors- FAQs

How do you find the common factors, what are common factors, what are the common factors of 3 and 5, what are the common factors of 10 and 15, what are the common factors of 12 and 15.

Put your understanding of this concept to test by answering a few MCQs. Click ‘Start Quiz’ to begin!

Select the correct answer and click on the “Finish” button Check your score and answers at the end of the quiz

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Factor mismatch measurement of resource-based cities: Evidence from all-round optimization total factor productivity decomposition framework

• Luo, Zhuoyang

In-depth analysis of the factors affecting the transformation of resource-based cities can provide effective support for the transformation and development of resource-dependent regions. How to comprehensively identify the factors affecting the transformation of resource-based cities is a complex problem. This study starts from the total factor productivity model and focuses on the two core basic factors that affect the transformation process of cities reliant on resources. Economic benefits and energy efficiency, respectively, from the economic benefit analysis framework and energy efficiency analysis framework for reconstruction, the two frameworks are combined with the use of distorted prices of resource elements to solve the problem that the synergistic effect of economic benefits and energy efficiency can not be measured. In order to quantitatively analyze the factors that affect the development efficiency of cities reliant on resources under the single or synergistic effect of the comprehensive framework, this study optimizes the directional distance function from three perspectives: exogenous weight, direction vector endogeneity, and absolute distance transformation relative distance, thus achieving an accurate assessment transformation efficiency of cities reliant on resources. Considering the impact of the new coronavirus epidemic, this study only selected the data of resource-based cities from 2003 to 2018, and found through model calculation that the impact on the transformation of cities reliant on resources: (1) Labor mismatch is mainly achieved by affecting the structure about the production of resource-based enterprises and industrial human resources; (2) Capital mismatch is mainly realized by affecting the production of resource-based enterprises; (3) Energy mismatch is mainly achieved by affecting high energy consumption enterprises and low production technology level enterprises. Further research shows that the main objects of these factors are the four parts of production technology level, energy consumption, total factor productivity and industrial structure. Through these contents, they affect environmental efficiency and deeply affect the transformation process of resource-based cities.

• Energy efficiency analysis framework;
• Economic benefit analysis framework;
• Resource-based city transformation;
• Optimization direction distance function

On solving a revised model of the nonnegative matrix factorization problem by the modified adaptive versions of the Dai–Liao method

• Original Paper
• Open access
• Published: 24 July 2024

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• Saman Babaie-Kafaki 1 ,
• Fatemeh Dargahi 2 &
• Zohre Aminifard 3  

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We suggest a revised form of a classic measure function to be employed in the optimization model of the nonnegative matrix factorization problem. More exactly, using sparse matrix approximations, the revision term is embedded to the model for penalizing the ill-conditioning in the computational trajectory to obtain the factorization elements. Then, as an extension of the Euclidean norm, we employ the ellipsoid norm to gain adaptive formulas for the Dai–Liao parameter in a least-squares framework. In essence, the parametric choices here are obtained by pushing the Dai–Liao direction to the direction of a well-functioning three-term conjugate gradient algorithm. In our scheme, the well-known BFGS and DFP quasi–Newton updating formulas are used to characterize the positive definite matrix factor of the ellipsoid norm. To see at what level our model revisions as well as our algorithmic modifications are effective, we seek some numerical evidence by conducting classic computational tests and assessing the outputs as well. As reported, the results weigh enough value on our analytical efforts.

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Avoid common mistakes on your manuscript.

1 Introduction

A cursory readout of the literature confirms that high-dimensional models have increasingly appeared in the data mining procedures, in the current age of social networks, bioinformatics, digital communications, and quantum computing. This fact places great importance on the necessity of diversifying the strategies for managing the difficulties that need to be prevailed when working with the complex, massive data sets.

A well-known plan to handle the high-dimensional models has been mainly centered on the compact representation of the input data sets [ 16 ]. In this regard, data reduction principally targets decreasing the size of the data sets while maintaining the important information, sometimes by data encoding procedures [ 20 ]. Meanwhile, when the data sets are given in the matrix forms, classic tools of the linear algebra such as nonnegative matrix factorization (NMF) may be greatly and influentially helpful [ 10 , 17 ]. As known, a wide range of the real-world data sets are inherently nonnegative and so, we should technically try to rule out the generation of the negative entries while managing and processing such data. Nowadays, NMF is repeatedly and purposefully applied in practical studies such as pattern recognition [ 11 ], recommendation systems [ 21 ] and face detection [ 29 ].

In a common framework, various NMF techniques take a matrix with nonnegative entries as the input, and deliver two lower dimension matrices with nonnegative entries as the output [ 16 ], in a way that multiplying the output matrices yields an accurate approximation for the input matrix. As a matter of fact, well-conditioning the intermediary consecutive approximations of the factorization elements may influentially enhance the computational stability [ 30 ], and as a result, make it possible to gain more appropriate output matrices as well.

Researchers have recently also pushed to devise memoryless versions of the classic algorithms as another move to handle the high-dimensional optimization models. To contrive a memoryless technique for a general minimization model, we should tactfully benefit the differential features of the cost function as well as the constraints. Meanwhile, the algorithmic steps should be simply performed, not being so time-consuming and labor-intensive, alongside keeping the accuracy at an acceptable level and ensuring the convergence of the solution trajectory. These features can be aggregately seen in the conjugate gradient (CG) algorithms which have been traditionally shaped in the vector forms [ 28 ]. Especially, the Dai–Liao (DL) method is nowadays labeled as an efficient CG algorithm due to flexibly incorporating the conjugacy and the quasi–Newton aspects in general circumstances [ 8 , 13 ].

Here, we plan to address possible model revisions as well as algorithmic modifications of some classic strategies for managing the large-scale data sets. To summarize the organization of our study, firstly we deal with a revised form of the classic measure function proposed by Dennis and Wolkowicz [ 14 ] in Section 2 , to be embedded to the optimization model of the NMF problem, by penalizing the ill-conditioned intermediary approximations of the factorization elements. Then, in Section 3 , we focus on determining adaptive formulas for the DL parameter as the solutions of a least-squares model formulated based on the ellipsoid vector norm [ 28 ]. We carry out numerical tests to mirror the value of our theoretical efforts in Section 4 , on the CUTEr problems [ 18 ] as well as a set of randomly generated NMF cases. Finally, we summarize some results for better understanding of the progress level in Section 5 .

2 A revised model for the nonnegative matrix factorization problem

Dimensionality reduction methodologies are naturally understood as influential approaches for analyzing large data sets. As known, high-dimensional data analysis is an integral part of the digital era due to recent developments in sensor technology. As mentioned in Section 1 , NMF is one such techniques that has caught researchers’ imagination thanks to the interpretability, simplicity, flexibility and generality [ 11 , 21 , 24 , 27 , 29 ].

Extracting hidden and important features from data gives rise to the NMF popularity in which the data matrix is approximated by the product of two matrices, usually much smaller than the original data matrix. All the input and output matrices of NMF (often) should be component wisely nonnegative. Mathematically speaking, for a given component wisely nonnegative matrix \(\textbf{A}\in \mathbb {R}^{m\times n}\) (or \(\textbf{A}\ge 0\) for short) and a positive integer \(r\ll \min \{m,n\}\) , NMF entails finding component wisely nonnegative matrices \(\textbf{W}\in \mathbb {R}^{m\times r}\) and \(\textbf{Z}\in \mathbb {R}^{r\times n}\) (or \(\textbf{W}\ge 0\) and \(\textbf{Z}\ge 0\) for short), by solving the following minimization problem:

where \(\Vert .\Vert _F\) stands for the Frobenius norm. In an efficient approach to address ( 2.1 ), the alternating nonnegative least-squares (ANLS) technique targets the following two subproblems [ 22 ]:

for all \(k\in \mathbb {Z}^+=\mathbb {N}\bigcup \{0\}=\{0,1,2,\dots \}\) .

As known, in the computational and analytical studies of the matrix spaces, a great deal of concern is devoted to the matrix condition number, an influential factor that is in a straight connection with the collinearity between the rows or the columns of the matrix [ 30 ]. Experiential efforts of the literature show that ill-conditioning may significantly deflect the solution process and yield misleading results. So, it is a classic matter of routine to devise a plan for having control over the condition number of the matrices that iteratively generate in an algorithmic procedure.

A cursory glimpse of the NMF literature shows a lack of analytical will as well as structural tendency to dealing with well-conditioning of the NMF outputs. It should be noted that various modified NMF models mainly target the orthogonality or symmetrization of the decomposition elements [ 17 ], being helpful in special applications of the data mining such as sparse recovery and clustering. Such extensions of the classic NMF model have been devised by imposing extra constraints to push the solution path toward the desired outputs. As a results, the solution process of the mentioned models can be to some extent challenging and sometimes, the workload may get heavy.

To depict the effect of ill-conditioning on the NMF model, here we report the outputs of the MATLAB function ‘nnmf’ on the well-known Hilbert matrix. Defined by

the Hilbert matrix \(\mathcal {H}\in \mathbb {R}^{n\times n}\) has been classically recognized as an ill-conditioned matrix, being also (symmetric) positive definite. By setting \(n=20\) and \(r=6\) , and then investigating the NMF outputs on \(\mathcal {H}\) obtained by 10000 different implementations of the MATLAB function ‘nnmf’, we observed that for more than 46% of the implementations, at least three columns (and rows) of \(\textbf{W}\) (and \(\textbf{Z}\) ) were equal to zero. That means for more than 46% of the implementations the outputs for \(r=4,5,6\) were quite the same. So, in such situations, the NMF cannot serve as a reliable tool in a recommender system for which filling the zero entries (empty positions) is of great importance. On the other hand, we observed that for at least 34% of the outputs the relative error was more than one. These observations could motivate us to deal with collinearity in the NMF model.

Combating the collinearity between the columns of \(\textbf{W}\) or the rows of \(\textbf{Z}\) , in order to take computational stability attitude toward the NMF model prompted us to plug condition number of the matrices \(\mathcal {W}=\textbf{W}^T\textbf{W}\) and \(\mathcal {Z}=\textbf{Z}\textbf{Z}^T\) of the dimension \(r\times r\) into the model ( 2.1 ). Note that the existence of sufficient (numerical) linear independency between the columns of \(\textbf{W}\) or the rows of \(\textbf{Z}\) , makes the matrices \(\mathcal {W}\) and \(\mathcal {Z}\) acceptably well-conditioned and positive definite. While, the mentioned collinearity pushes \(\mathcal {W}\) and \(\mathcal {Z}\) toward ill-conditioning and positive semidefiniteness. So, to be cautious about such troubling issues, the following revised version of the NMF model ( 2.1 ) can be proffered:

where \(\lambda _1\ge 0\) and \(\lambda _2\ge 0\) are the penalty parameters [ 25 , 26 ] and the maximum magnification (maxmag) and the minimum magnification (minmag) by an arbitrary matrix \(P\in \mathbb {R}^{m\times n}\) are respectively defined in Watkins [ 30 ] as

As seen, ill-conditioned choices for \(\textbf{W}\) and \(\textbf{Z}\) meaningfully impose penalty to the model. Meanwhile, although seldom occurs in practice, \(\mathfrak {\hat{F}}(\textbf{W},\textbf{Z})\) is not well-defined when \(\textbf{W}\) or \(\textbf{Z}\) are rank deficient.

In the model ( 2.4 ) well-conditioning has been brought up by straightly embedding penalty terms to the cost function. So, in this respect, since we made the solution process away from the possible troubling consequences resulted by imposing an extra set of constraints, finding approximate solutions of the model may be less challenging. However, we should not overlook the complexity of doing computations by the spectral condition number in the model, especially in large-scale cases. It is generally a matter of fact that carrying out calculations with high-dimensional dense matrices causes extra CPU time and may increase the numerical errors as well. So, developing sparse approximations of such matrices in the data analysis has recently attracted special attentions [ 25 , 26 ].

Among the fundamental sparse structures for the symmetric matrices, there exist the diagonal and the (banded) symmetric tridiagonal matrices [ 7 ] as well as the symmetric rank-one or rank-two updates of the (scaled) identity matrix [ 28 ]. In essence, we should conduct a cost-benefit analysis to select a special sparse matrix structure which is of enough suitability in the relevant application. Driven by this issue, because of the presence of the spectral condition number in the augmented model ( 2.4 ) which is directly linked to the eigenvalues of the matrix, to tackle some precarious situations stemming from a great deal of time-consuming for calculating \(\mathcal {W}\) and \(\mathcal {Z}\) , it may be preferable to use diagonal approximations of \(\mathcal {W}\) and \(\mathcal {Z}\) in the model ( 2.4 ) by

Notably, the above diagonal estimations are derived from

where \(\textbf{D}^+\) denotes the collection of all diagonal matrices with the nonnegative elements in \(\mathbb {R}^{r\times r}\) .

As known, measure functions provide helpful tools to evaluate and analyze well-conditioning of the square matrices. They often target the distribution of the matrix eigenvalues [ 25 ]. Among them, as a fundamental study to analyze the scaling and sizing of the quasi–Newton updates, Dennis and Wolkowicz [ 14 ] proposed the following measure function:

for an arbitrary positive definite matrix \(\textbf{A}\in \mathbb {R}^{r\times r}\) . As a factor to evaluate well-conditioning, \(\psi (\textbf{A})\) considers all the eigenvalues of \(\textbf{A}\) , rather than, as occurs in the spectral condition number, only taking the extreme eigenvalues of the matrix [ 30 ]. So, by employing \(\psi (.)\) instead of \(\kappa (.)\) in ( 2.4 ), it is more likely possible to obtain NMF elements with well-distributed eigenvalues. However, the matrix function ( 2.5 ) would be accompanied by some kinds of complexity due to its denominator.

Mathematical inequalities have been widely and purposefully employed by the researchers to turn a dense or complicated formula into something manageable. For this aim, the first and foremost point in accordance with the norm of the literature is to rise the level of interpretability of the targeted formula or model. Here, for the sake of a well-planned simplicity that is a crucial issue in the high-dimensional data analysis, we organize assistance from the first part of the mean inequality chain that is related to the algebraic ties between the harmonic, geometric, arithmetic, and quadratic means. To proceed, firstly note that \(\text {det}(\textbf{A})=\displaystyle \prod _{i=1}^{r}\zeta _i\) , in which \(\{\zeta _i\}_{i=1}^r\) is the set of the eigenvalues of \(\textbf{A}\) . Therefore, bearing the relation between the geometric and the harmonic means in mind, here in the sense of

and noting that the trace of a (square) matrix is equal to the sum of its eigenvalues, the following simple bound for \(\psi (\textbf{A})\) can be obtained:

This gives rise compelling motivations to employ \(\varphi (.)\) instead of \(\kappa (.)\) in ( 2.4 ), to possibly gain NMF elements with well-distributed eigenvalues. So, the modified model is given by

Inherited from the measure function ( 2.5 ), the penalty terms of the model ( 2.6 ) control the condition number by engaging in all the diagonal elements of the relevant matrices, not only considering the extreme ones. Also, emerging polynomial terms makes the model easier to handle with respect to determining the gradient of the cost function.

The major defect of the cost function of the model ( 2.6 ) is that it is not differentiable everywhere due to the extra penalty terms. Especially, if \(\textbf{W}\) has a zero column or \(\textbf{Z}\) has a zero row, then \(\mathfrak {\breve{F}}\) in ( 2.6 ) is not well-defined. Moreover, small magnitudes of the columns of \(\textbf{W}\) or the rows of \(\textbf{Z}\) are computationally troublesome. Backed by these arguments and in favor of simplicity, our revised ANLS (RANLS) method is founded upon the following modified version of the model ( 2.6 ):

with some constant \(\gamma >0\) . As seen, \(\mathfrak {\tilde{F}}\) is well-defined and also, it is differentiable everywhere. Thus, the next revised versions of the least-squares models ( 2.2 ) and ( 2.3 ) should alternately be solved:

for all \(k\in \mathbb {Z}^+\) .

3 Adaptive optimal choices for the Dai–Liao parameter based on the ellipsoid norm

Among the fundamental techniques for solving the unconstrained minimization problem \(\displaystyle \min _{x\in \mathbb {R}^n} f(x)\) , the CG methods are iteratively defined by

starting by some \(x_0\in \mathbb {R}^n\) and \(d_0=-g_0\) , in which \(g_k=\triangledown f(x_k)\) and \(\beta _k\in \mathbb {R}\) is the CG parameter [ 3 ]. Also, the scalar \(\alpha _k>0\) , called the step length, is customarily determined as the output of an approximate line search, popularly to meet the (strong) Wolfe conditions [ 28 ]. Here, we assume that the cost function f is smooth and its gradient is analytically available. Also, \(\Vert .\Vert \) signifies the \(\ell _2\) (Euclidean) norm and our analysis undergoes with the Wolfe conditions for which \(s_k^Ty_k>0\) , where \(s_k=x_{k+1}-x_k=\alpha _k d_k\) .

In the initial years of the current century, the worthy study of Dai and Liao [ 13 ] brought considerable attention to the CG techniques in various guidelines [ 4 ]. Recently, Babaie–Kafaki [ 8 ] conducted an expository review on the DL method to provide a better understanding of the capabilities of the method from several standpoints. For the DL method, \(\beta _k\) in its original form is set to

with \(y_k=g_{k+1}-g_k\) , where the scalar \(t>0\) is called the DL parameter. It is valuable to note that if

then the DL directions satisfy the sufficient descent condition which is an important ingredient of the convergence [ 5 ].

Among the analytical attempts to seek an appropriate formula for t as a classic open problem [ 4 , 8 ], Babaie–Kafaki and Ghanbri [ 9 ] offered a least-squares model, i.e.

by pushing the DL direction to the direction of the three-term CG method proposed by Zhang, Zhou and Li (ZZL) [ 31 ]. As known, the ZZL directions satisfy a strong form of the sufficient descent condition. Moreover, they benefit the consecutive gradient differences vector \(y_k\) as an element of the search direction, besides the vectors \(g_{k+1}\) and \(d_k\) in the framework of a linear combination, rather than the DL directions that are just linear combination of \(g_{k+1}\) and \(d_k\) . As a result of their plan, Babaie–Kafaki and Ghanbri [ 9 ] obtained the following formula for t :

Here, we organize assistance from the ellipsoid vector norm to diversify the adaptive choices for the DL parameter. As an extended form of the \(\ell _2\) norm in the sense of

where \(\mathcal {M}\in \mathbb {R}^{n\times n}\) is a (symmetric) positive definite matrix, ellipsoid norm has been pivotally employed to analyze the convergence of the steepest descent and the quasi–Newton methods, and particularly, to devise the scaled trust region algorithms [ 28 ]. In our strategy, we plan to set several choices for \(\mathcal {M}\) using the quasi–Newton updating formulas [ 28 ].

Quasi–Newton methods have been traditionally devised to tactfully estimate the (inverse) Hessian in order to determine the search direction in the iterative continuous optimization techniques. Mostly being positive definite, the given matrix approximations classically should satisfy the (standard) secant equation, i.e. \(\textbf{B}_{k+1}s_k=y_k\) , where \(\textbf{B}_{k+1}\approx \nabla ^2f(x_{k+1})\) [ 3 , 28 ]. The methods benefit enough flexibility to effectively address the large-scale models. For this aim, the memoryless versions of the well-known BFGS and DFP quasi–Newton updating formulas can be applied [ 6 ]; that is,

both are positive definite approximations of \(\nabla ^2f(x_{k+1})^{-1}\) , for all \(k\in \mathbb {Z}^+\) . Here, MLBFGS and MLDFP are respectively shortened forms of the ‘memoryless BFGS’ and the ‘memoryless DFP’.

It is a matter of tradition that reform is always needed in the available algorithms to answer the great need of diversity and inclusion. There are a lot of evidence in the methodology literature that such efforts evolved the algorithmic schemes over time. So, we should not neglect effects of these evolutionary plans on the hybrid CG algorithms as well. By this fact at the forefront, here we consider the ellipsoid extension of the least-squares model ( 3.4 ) as follows:

which yields

So, \(t_k^{\text {ZZL}}\) given by ( 3.5 ) is the solution of ( 3.4 ) by setting \(\mathcal {M}\) as the identity matrix. Also, if we let \(\mathcal {M}=\textbf{B}_{k+1}\) given by a quasi–Newton update for the Hessian, then, because of the standard secant equation we have

which is an effective formula already suggested by Dai and Kou (DK) [ 12 ]. This salient fact places great importance on the effectiveness of the given extended least-squares model. The setting \(t=t_k^{\text {DK}}\) in the DL method ensures ( 3.3 ) that squarely leads to the sufficient descent property. Moreover, if we let \(\mathcal {M}=\textbf{H}_{k+1}^{\text {MLBFGS}}\) or \(\mathcal {M}=\textbf{H}_{k+1}^{\text {MLDFP}}\) , then we respectively obtain

From the Cauchy–Schwarz inequality, it can be seen that \(t_k^{\text {MLBFGS}}>0\) and \(t_k^{\text {MLDFP}}>0\) . To gain the sufficient descent property in light of ( 3.3 ), here we propose the following restricted versions of ( 3.7 ) and ( 3.8 ):

As a result, global convergence of the DL method with the given formulas for t can be proved following the analysis of [ 2 , 13 ].

4 Computational experiments

We offer here some computational confirmation for the veracity of our theoretical analyses, starting with some numerical tests on the CUTEr library [ 18 ] with \(n\ge 50\) , comprising of 96 problems. All the tests were performed by MATLAB version 7.14.0.739 (R2012a), installed on the Centos 6.2 server Linux operation system, in a computer AMD FX–9800P RADEON R7 with 12 COMPUTE CORES 4C+8G 2.70 GHz of CPU and 8 GB of RAM. The effectuality of the parametric choices ( 3.6 ), ( 3.9 ), ( 3.10 ) and the Hager–Zhang (HZ) formula [ 19 ], i.e.

is appraised for the DL+ method with

proposed for establishing convergence for general cost functions [ 13 ]. In our tests, DK+, DL–BFGS+, DL–DFP+ and HZ+, stand for the iterative method ( 3.1 ) with the CG parameter ( 4.2 ), in which t is respectively computed by ( 3.6 ), ( 3.7 ), ( 3.8 ) and ( 4.1 ). Since in rare iterations the DL+ method may fail to generate descent direction, restart (by the negative gradient vector) has been also employed as suggested in Dai and Liao [ 13 ].

For the algorithms, we used the approximate Wolfe conditions of Hager and Zhang [ 19 ] with the similar settings, and let the stopping criteria as \(k>10000\) or \(\Vert g_k\Vert <10^{-6}(1+|f_k|)\) . Also, we set \(\eta =0.26\) in ( 3.9 ) and ( 3.10 ), to enhance the possibility of employing ( 3.7 ) and ( 3.8 ). To visually assess the algorithmic results, we applied the Dolan–Moré performance profile [ 15 ], by comparisons based on the TNFGE and CPUT metrics, being acronyms for the ‘total number of function and gradient evaluations’ (as outlined in Hager and Zhang [ 19 ]) and the ‘CPU time’, respectively. Figure 1 represents the results, by which it can be seen that DL–BFGS+ and DL–DFP+ are generally preferable to DK+ and HZ+, especially with respect to TNFGE. Meanwhile, with respect to CPUT, at times DK+ and HZ+ are competitive with DL–BFGS+ and DL–DFP+. This observation is mainly related to the structure of the formulas ( 3.7 ) and ( 3.8 ) which is to some extent more complex rather than ( 3.6 ) and ( 4.1 ). Also, since DL–DFP+ is slightly preferable to DL–BFGS+, we can conclude that the setting ( 3.8 ) for the DL+ method is more effective than the setting ( 3.7 ).

Performance profile plots for DK+, DL–BFGS+, DL–DFP+ and HZ+ based on TNFGE (A) and CPUT (B)

To bring the validity of the given revised NMF model to light, in this part of our computational experiments we investigate the efficiency of DL–DFP+ for ANLS by solving the least-squares subproblem ( 2.2 )–( 2.3 ) of the minimization model ( 2.1 ), and for RANLS by solving the least-squares subproblem ( 2.8 )–( 2.9 ) of the minimization model ( 2.7 ). To handle the nonnegativity constraints in the subproblems, we followed the suggestion of Li et al. [ 22 ] and employed a proximal scheme in the sense of setting the negative entries of the iterative outputs equal to zero. For RALNS, we set \(\lambda _1=\lambda _2=1\) and \(\gamma =10^{-10}\) in ( 2.7 ), and for both ANLS and RANLS, we adopted the termination condition of Liu and Li [ 23 ] as well; which is

with \(\mathcal {F}=\mathfrak {F}\) and \(\mathcal {F}=\tilde{\mathfrak {F}}\) , respectively, and \(\nu =10^{-2}\) . By using the uniform distribution, the test matrices were generated randomly with various dimensions, together with the initial estimates of the NMF elements, as declared in Ahookhosh et al. [ 1 ]. Outputs have been outlined in Table 1 , including the spectral condition number (Cond) and the relative error (RelErr), calculated by

To recapitulate the results, we can observe that RANLS and ANLS are approximately competitive with respect to the accuracy. While, in the condition number viewpoint which is the main target of this study, RANLS is generally preferable to ANLS. Hence, capability of delivering well-conditioned NMF elements with satisfactory accuracy can therefore be considered a success by RANLS.

5 Conclusions

We have mainly concentrated on the modifying a classic optimization model of the nonnegative matrix factorization problem, frequently arising in a wide range of practical fields. Avoiding the possibility of ill-conditioning in the results of the decomposition motivated us to revise the model by embedding a measurement for condition numbers of the diagonalized types of the output matrices. What embedded as the well-conditioner (penalty) term has been extracted from the Dennis–Wolkowicz measure function [ 14 ]. Then, based on an ellipsoid norm-oriented least-squares model, some optimal choices for the Dai–Liao parameter have been suggested. Driven by the great need for algorithmic tools with the low memory consumption of the machine, the ellipsoid norms have been centered on the memoryless BFGS and DFP formulas. The approach in terms of which the method’s influential parameter has been computed is tending the Dai–Liao search direction to that of a well-functioning three-term conjugate gradient algorithm. Then, to examine the performance of the Dai–Liao method when it is equipped with the given formulas as the parametric settings, some computational tests were performed on the CUTEr functions. The findings were evaluated leveraged on the well-known Dolan–Moré benchmark. The results demonstrated the positive impact of our suggestions for the Dai–Liao parameter. Furthermore, the quality of the given revised nonnegative matrix factorization model has been assessed in several random cases. The results showed that the revised model can produce more well-conditioned factorization elements with reasonable relative errors. Thus, in practical terms, computational experiments have supported our theoretical assertions.

Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Ahookhosh, M., Hien, L.T.K., Gillis, N., Patrinos, P.: Multi-block Bregman proximal alternating linearized minimization and its application to orthogonal nonnegative matrix factorization. Comput. Optim. Appl. 79 (3), 681–715 (2021)

Article   MathSciNet   Google Scholar  

Aminifard, Z., Babaie-Kafaki, S.: An optimal parameter choice for the Dai–Liao family of conjugate gradient methods by avoiding a direction of the maximum magnification by the search direction matrix. 4OR. 17 , 317–330 (2019)

Andrei, N.: Modern Numerical Nonlinear Optimization. Switzerland, Springer, Cham (2006)

Google Scholar  

Andrei, N.: Open problems in conjugate gradient algorithms for unconstrained optimization. B. Malays. Math. Sci. So. 34 (2), 319–330 (2011)

MathSciNet   Google Scholar  

Babaie-Kafaki, S.: On the sufficient descent condition of the Hager–Zhang conjugate gradient methods. 4OR. 12 (3), 285–292 (2014)

Babaie-Kafaki, S.: On optimality of the parameters of self-scaling memoryless quasi-Newton updating formulae. J. Optim. Theory Appl. 167 (1), 91–101 (2015)

Babaie-Kafaki, S.: A monotone preconditioned gradient method based on a banded tridiagonal inverse Hessian approximation. UPB Sci. Bull. Ser. A: Appl. Math. Phys. 80 (1), 55–62 (2018)

Babaie-Kafaki, S.: A survey on the Dai-Liao family of nonlinear conjugate gradient methods. RAIRO-Oper. Res. 57 , 43–58 (2023)

Babaie-Kafaki, S., Ghanbari, R.: Two optimal Dai-Liao conjugate gradient methods. Optimization 64 (11), 2277–2287 (2015)

Berry, M.W., Browne, M., Langville, A.N., Pauca, V.P., Plemmons, R.J.: Algorithms and applications for approximate nonnegative matrix factorization. Comput. Stat. Data Anal. 52 (1), 155–173 (2007)

Cho, Y.C., Choi, S.: Nonnegative features of spectro-temporal sounds for classification. Pattern Recognit. Lett. 26 (9), 1327–1336 (2005)

Article   Google Scholar  

Dai, Y.H., Kou, C.X.: A nonlinear conjugate gradient algorithm with an optimal property and an improved Wolfe line search. SIAM J. Optim. 23 (1), 296–320 (2013)

Dai, Y.H., Liao, L.Z.: New conjugacy conditions and related nonlinear conjugate gradient methods. Appl. Math. Optim. 43 (1), 87–101 (2001)

Dennis, J.E., Wolkowicz, H.: Sizing and least-change secant methods. SIAM J. Numer. Anal. 30 (5), 1291–1314 (1993)

Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91 (2, Ser. A), 201–213 (2002)

Eldén, L.: Matrix Methods in Data Mining and Pattern Recognition. SIAM, Philadelphia (2007)

Book   Google Scholar  

Gan, J., Liu, T., Li, L., Zhang, J.: Nonnegative matrix factorization: a survey. Comput. J. 64 (7), 1080–1092 (2021)

Gould, N.I.M., Orban, D., Toint, Ph.L.: CUTEr: a constrained and unconstrained testing environment, revisited. ACM Trans. Math. Software 29 (4), 373–394 (2003)

Hager, W.W., Zhang, H.: Algorithm 851: CG \(_{-}\) Descent, a conjugate gradient method with guaranteed descent. ACM Trans. Math. Software 32 (1), 113–137 (2006)

Han, J., Kamber, M., Pei, J.: Data Mining: Concepts and Techniques, 3rd Edition. Morgan Kaufmann (2012)

Koren, Y., Bell, R., Volinsky, C.: Matrix factorization techniques for recommender systems. Computer 42 (8), 30–37 (2009)

Li, X., Zhang, W., Dong, X.: A class of modified FR conjugate gradient method and applications to nonnegative matrix factorization. Comput. Math. Appl. 73 , 270–276 (2017)

Liu, H., Li, X.: Modified subspace Barzilai-Borwein gradient method for non-negative matrix factorization. Comput. Optim. Appl. 55 (1), 173–196 (2013)

Pompili, F., Gillis, N., Absil, P.A., Glineur, F.: Two algorithms for orthogonal nonnegative matrix factorization with application to clustering. Neurocomputing 141 , 15–25 (2014)

Roozbeh, M., Babaie-Kafaki, S., Aminifard, Z.: Two penalized mixed-integer nonlinear programming approaches to tackle multicollinearity and outliers effects in linear regression models. J. Ind. Manag. Optim. 17 (6), 3475 (2021)

Roozbeh, M., Babaie-Kafaki, S., Aminifard, Z.: Improved high-dimensional regression models with matrix approximations applied to the comparative case studies with support vector machines. Optim. Methods Softw. 37 (5), 1912–1929 (2022)

Shahnaz, F., Berry, M.W., Pauca, V.P., Plemmons, R.J.: Document clustering using nonnegative matrix factorization. Inf. Process. Manag. 42 (2), 373–386 (2006)

Sun, W., Yuan, Y.X.: Optimization Theory and Methods: Nonlinear Programming. Springer, New York (2006)

Wang, Y., Jia, Y., Hu, C., Turk, M.: Nonnegative matrix factorization framework for face recognition. Int. J. Pattern Recognition Artif. Intell. 19 (04), 495–511 (2005)

Watkins, D.S.: Fundamentals of Matrix Computations. John Wiley and Sons, New York (2002)

Zhang, L., Zhou, W., Li, D.H.: Some descent three-term conjugate gradient methods and their global convergence. Optim. Methods Softw. 22 (4), 697–711 (2007)

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The authors thank the anonymous reviewers for their worthy comments helped to improve the quality and organization of this work.

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Babaie-Kafaki, S., Dargahi, F. & Aminifard, Z. On solving a revised model of the nonnegative matrix factorization problem by the modified adaptive versions of the Dai–Liao method. Numer Algor (2024). https://doi.org/10.1007/s11075-024-01886-w

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DOI : https://doi.org/10.1007/s11075-024-01886-w

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• Unconstrained optimization
• Nonnegative matrix factorization
• Conjugate gradient algorithm
• Quasi–Newton updating formula
• Ellipsoid norm

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