Factors

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GCSE Maths Number FMP Factors And Multiples

Factors

Here we will learn about factors, including recognising factors, commutativity, how to systematically find all factor pairs of a number, and solving problems using factors.

There are also factors 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?

Factors are numbers that will divide into an integer (a whole number) with no remainder.

Factors are always integers and can sometimes be called divisors. Every integer has at least $\bf{2}$ factors.

If an integer has only two factors, it is a prime number.

For example, the factors of $5$ are $1$ and $5.$

Step-by-step guide: Prime numbers

If an integer has an odd number of factors, it is a square number.

For example, the factors of $4$ are $1, 2,$ and $4.$ This is because one factor is repeated.

Step-by-step guide: Square numbers

A factor pair consists of two numbers that are factors of another number and make that number when they are multiplied together.

For example, the factor pairs of $12$ are,

\begin{aligned} &1\times{12}\\\\ &2\times{6}\\\\ &3\times{4} \end{aligned}

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

Factor pairs have a commutative property such that you can switch the order of the calculation and the calculation remains the same: $2\times{3}=3\times{2}.$

Prime factors are factors of a number that are also prime numbers.

For example, the prime factors of $30=2\times{3}\times{5}.$

Step-by-step guide: Prime factors

Once we know what the factors of a number are, we can use these to solve problems involving probability, substitution, completing tables, and area.

We use factors in other areas of maths including factorising algebraic expressions and quadratics, simplifying and equivalent fractions, calculating the highest common factor for a pair of integers, area calculations, and enlargement.

What are factors?

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 worksheet

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

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Factors worksheet

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

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Related lessons on factors and multiples

Factors is part of our series of lessons to support revision on factors and multiples and factors, multiples and primes. You may find it helpful to start with the main factors and multiples 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 examples

Example 1: listing factors (even number)

List the factors of $24.$

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

As $n=24,$ we have the first factor pair $1\times{24}.$

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

As $24$ is an even number, $2$ is a factor of $24.$

24\div{2}=12

so the next factor pair is $2\times{12}$ .

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

So far we have:

\begin{aligned} &1\times{24}\\\\ &2\times{12}\\ \end{aligned}

24\div{3}=8

and so we have the next factor pair $3\times{8}$ .

24\div{4}=6

and so we have the next factor pair $4\times{6}$ .

24\div{5}=4.8

which is a decimal so $5$ is not a factor of $24$ .

The next factor to try is $6.$ As factors are commutative,

6\times{4}=4\times{6}

which is the same as the previous factor pair.

We have now found all of the factor pairs:

\begin{aligned} &1\times{24}\\\\ &2\times{12}\\\\ &3\times{8}\\\\ &4\times{6} \end{aligned}

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

Reading down the first column of factors, and up the second column, the factors of $24$ are: $1, 2, 3, 4, 6, 8, 12, 24.$

Example 2: listing factors (odd number)

List the factors of $27.$

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

As $n=27,$ we have the first factor pair $1\times{27}.$

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

As $27$ is an odd number, $2$ is not a factor of $27.$

However, $27\div{3}=9$

and so the next factor pair is $3\times{9}.$

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

So far we have:

$\begin{aligned} &1\times{27}\\\\ &3\times{9} \end{aligned}$

$27$ has a remainder when it is divided by $4, 5, 6, 7,$ or $8.$

The next factor to try is $9,$ but we already know that $3\times{9}$ is a factor pair and this is the same as $9\times{3}.$

We have now found all of the factor pairs:

$\begin{aligned} &1\times{27}\\\\ &3\times{9} \end{aligned}$

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

Reading down the first column of factors, and up the second column, the factors of $27$ are: $1, 3, 9,$ and $27.$

Example 3: listing factors (square number)

List the factors of $36.$

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

As $n=36,$ we have the first factor pair $1\times{36}.$

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

As $36$ is an even number, $2$ is a factor of $36.$

$36\div{2}=18$

and so the next factor pair is $2\times{18}.$

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

So far we have:

$\begin{aligned} &1\times{36}\\\\ &2\times{18} \end{aligned}$

$36\div{3}=12$

and so the next factor pair is $3\times{12}$ .

$36\div{4}=9$

and so the next factor pair is $4\times{9}$ .

$36$ has a remainder when it is divided by $5$ and so $5$ is not a factor.

$36\div{6}=6$

and so the next factor pair is $6\times{6}$ .

We have reached a repeated factor and so we have found all of the factor pairs for $36:$

$\begin{aligned} &1\times{36}\\\\ &2\times{18}\\\\ &3\times{12}\\\\ &4\times{9}\\\\ &6\times{6} \end{aligned}$

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

Reading down the first column of factors, and up the second column, the factors of $36$ are: $1, 2, 3, 4, 6, 9, 12, 18,$ and $36.$

Notice that as we have a repeated factor, we have an odd number of factors in the list above. This can help us determine that $36$ is a square number.

Example 4: listing factors (prime number)

List the factors of $7.$

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

As $n=7,$ we have the first factor pair $1\times{7}.$

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

As $7$ is an odd number, $2$ is not a factor of $7.$

$7$ has a remainder when it is divided by $2, 3, 4, 5,$ and $6.$

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

The next integer to try is $7,$ which we have already used

$(1\times{7}=7\times{1})$

and so the only factor pair for the number $7$ is $1\times{7}.$

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

The factors of $7$ are: $1$ and $7.$

As $7$ has only two factors, $1$ and itself, $7$ is a prime number.

Example 5: listing factors (3 digit number)

List the factors of $200.$

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

As $n=200,$ we have the first factor pair $1\times{200}.$

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

As $200$ is an even number, $2$ is a factor of $200.$

$200\div{2}=100$

and so the next factor pair is $2\times{100}.$

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

So far we have:

$\begin{aligned} &1\times{200}\\\\ &2\times{100} \end{aligned}$

$200$ has a remainder when it is divided by $3$ and so $3$ is not a factor.

$200\div{4}=50$

and so the next factor pair is $4\times{50}.$

$200\div{5}=40$

and so the next factor pair is $5\times{40}.$

$200$ has a remainder when it is divided by $6$ and $7$ and so these are not factors of $200.$

$200\div{8}=25$

and so the next factor pair is $8\times{25}.$

$200$ has a remainder when it is divided by $9$ and so $9$ is not a factor.

$200\div{10}=20$

and so the next factor pair is $10\times{20}$ .

$200$ has a remainder when it is divided by $11, 12, 13, 14, 15, 16, 17, 18,$ and $19$ and so these are not factors of $200.$

The next integer to try is $20$ but this appears in the previous factor pair

$(10\times{20}=20\times{10}).$

We have therefore found all of the factor pairs for $200:$

$\begin{aligned} &1\times{200}\\\\ &2\times{100}\\\\ &4\times{50}\\\\ &5\times{40}\\\\ &8\times{25}\\\\ &10\times{20} \end{aligned}$

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

Reading down the first column of factors, and up the second column, the factors of $200$ are: $1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100,$ and $200.$

Example 6: listing factors (common factors)

The factors of $6$ are $1, 2, 3,$ and $6.$ By calculating the factors of $8,$ determine the common factors of $6$ and $8.$

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

As we want to list the common factors of $6$ and $8,$ we need to find the factors of each of them, and then highlight common factors (the numbers that appear in both lists).

We already have the factors of $6,$ so we just need to find the factor pairs for $8.$

The first factor pair is $1\times{8}$ .

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

As $8$ is an even number, $2$ is a factor of $8.$

$8\div{2}=4$

and so the next factor pair is $2\times{4}.$

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

8 has a remainder when it is divided by 3.

The next factor to try is 4 and we have already used this in the previous factor pair $(2\times{4}=4\times{2}).$

We have found all of the factor pairs:

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

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

The factors of $8$ are: $1, 2, 4,$ and $8.$

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

The common factors of $6$ and $8$ are: $1$ and $2.$

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.

Practice factors questions

12, 24, 36, 48, 60

$1$ and $12, 2$ and $6$

2\times{2}\times{3}

1, 2, 3, 4, 6, 12

The factor pairs of $12$ are:

\begin{aligned} &1\times{12}\\\\ &2\times{6}\\\\ &3\times{4} \end{aligned}

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

1, 3, 5, 15

15, 30, 45, 60, 75, 90

$3$ and $5$

3\times{5}

The factor pairs of $15$ are:

\begin{aligned} &1\times{15}\\\\ &3\times{5} \end{aligned}

So the factors of $15$ are $1, 3, 5$ and $15.$

1, 2, 4, 8, 16, 32, 64

$1$ and $64, 2$ and $32, 8$

64, 128, 192, 256, 320

2\times{2}\times{2}\times{2}\times{2}\times{2}

The factor pairs of $64$ are:

\begin{aligned} &1\times{64}\\\\ &2\times{32}\\\\ &4\times{16}\\\\ &8\times{8} \end{aligned}

So the factors of $64$ are $1, 2, 4, 8, 16, 32,$ and $64.$

17, 34, 51, 68, 85, 102

17

$1$ and $17$

10+7

The factor pairs of $17$ are:

1\times{17}

So the factors of $17$ are $1$ and $17.$

2\times{2}\times{2}\times{3}\times{5}

$12$ and $10, 6$ and $20, 4$ and $30.$

1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60

120, 240, 360, 480, 600

The factor pairs of $120$ are:

\begin{aligned} &1\times{120}\\\\ &2\times{60}\\\\ &3\times{40}\\\\ &4\times{30}\\\\ &5\times{24}\\\\ &6\times{20}\\\\ &8\times{15}\\\\ &10\times{12} \end{aligned}

So the factors of $120$ are: $1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40,$ and $60.$

10, 20, 30, 40, 50

20, 40, 60, 80, 100

$1, 2, 5,$ and $10$

1\times{2}\times{5}

The factor pairs of $20$ are:

\begin{aligned} &1\times{20}\\\\ &2\times{10}\\\\ &4\times{5} \end{aligned}

So the factors of $20$ are: $1, 2, 4, 5, 10,$ and $20.$

The factors of $10$ are: $1, 2, 5,$ and $10.$

The common factors of $10$ and $20$ are: $1, 2, 5,$ and $10.$

Factors GCSE questions

1. Here is a list of numbers:

2, \quad 3, \quad 6, \quad 9, \quad 12, \quad 18, \quad 24

(a) Which numbers are factors of $18?$

(b) Which numbers have a common factor of $6?$

(d) Show that $3$ is a factor of $108.$

(e) Which number has an odd number of factors?

(5 marks)

Show answer

(a) $2, 3, 6, 9, 18$

(1)

(b) $6, 12, 18, 24$

(1)

(d) $108\div{3}=36$ and no remainder stated

(1)

(e) $9$

(1)

2. Let $x$ and $y$ be two single digit prime numbers where:

$xy$ is a multiple of $5$
$x > y.$

(a) List the possible values of $x.$

(b) Jamal says “ $21$ is a prime number as it only has $2$ factors, $1$ and $21$ ”. Is Jamal correct? Explain your answer.

(5 marks)

Show answer

(a)

5\times{3}=15\textbf{ and }7\times{5}=35

(1)

x=5\text{ and }7

(1)

(b)

(1)

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

(1)

$21$ has more than $2$ factors and so it is not prime.

(1)

3. A spinner is spun twice. The outcomes are multiplied and written into the sample space diagram below.

Factors gcse question 3

(a) Calculate the probability of getting a factor of $6$ after $2$ spins.

(b) What is the probability of getting a factor of $8,$ given that the first spin is a $2?$

(4 marks)

Show answer

(a)

Factors of $6: 1, 2, 3, 6$

(1)

\frac{7}{9}

(1)

(b)

Factors of $8: 1, 2, 4, 8$

(1)

\frac{2}{3}

(1)

4. The Venn diagram below shows the two sets A = Factors of $32,$ and B = Factors of $40$ with some values placed.

Factors gcse question 4

(a) Add all of the factors of $32$ and all of the factors of $40$ to the Venn diagram.

(b) I pick a factor at random. What is the probability that the factor is also a factor of $12?$

(5 marks)

Show answer

(a)

Factors of $32: 1, 2, 4, 8, 16, 32$

(1)

Factors of $40: 1, 2, 4, 5, 8, 10, 20, 40$

(1)

Completed Venn diagram:

Factors gcse question 4 answer

(1)

(b)

Factors of $12: 1, 2, 3, 4, 6, 12$

(1)

\frac{3}{10}

(1)

Learning checklist

You have now learned how to:

Use and understand the term factor

Recognise and use factor pairs and commutativity in mental calculations’

Identify factors including all factor pairs of a number and common factors of two numbers

Solve problems involving multiplying and dividing including knowledge of factors

The next lessons are

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Introduction

What are factors?

How to list factors

Factors worksheet

Factors examples

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Example 1: listing factors (even number) Example 2: listing factors (odd number) Example 3: listing factors (square number) Example 4: listing factors (prime number) Example 5: listing factors (3 digit number) Example 6: listing factors (common factors)

Common misconceptions

Practice factors questions

Factors GCSE questions

Learning checklist

Next lessons

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