Factors and multiples

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GCSE Maths Number Factors, Multiples And Primes

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.

What are factors and multiples?

Factors

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

Multiples

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.

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Factors and multiples worksheet

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

DOWNLOAD FREE

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 examples

Example 1: listing factors (odd number)

List the factors of $15.$

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

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

2Write 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 $3\times{5}.$

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

So far you have,

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

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,

\begin{aligned} &1\times{15}\\\\ &3\times{5} \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 $15$ are $1, 3, 5,$ and $15.$

Example 2: listing factors (square number)

List the factors of $16.$

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

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

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

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

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

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

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 $16,$

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

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

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 $21.$

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

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 $1\times{6}.$

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

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 $2\times{3}.$

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

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,

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

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

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 $12.$

State the first multiple of $\bf{n}$ .

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

Calculate the second multiple of $\bf{n}$ .

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

Continue until you have calculated the number of multiples needed.

\begin{aligned} &12\times{3}=36 \\\\ &12\times{4}=48 \\\\ &12\times{5}=60 \end{aligned}

Write the solution.

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 $6?$

State the first multiple of $\bf{n}$ .

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 $6.$

Calculate the second multiple of $\bf{n}$ .

Move on to step 3.

Continue until you have calculated the number of multiples needed.

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

Write the solution.

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 $12.$

State the first multiple of $\bf{n}$ .

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.

Calculate the second multiple of $\bf{n}$ .

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

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

Continue until you have calculated the number of multiples needed.

8\times{3}=24

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

Write the solution.

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

$2$ and $12, 3$ and $8, 4$ and $6$

$24, 48, 72, 96,$ and $120$

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

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

The factor pairs of $24$ are,

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

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

$1$ and $49$

49, 98, 147, 196, 245

$1, 7,$ and $49$

7\times{7}

The factor pairs of $49$ are,

\begin{aligned} &1\times{49}\\\\ &7\times{7} \end{aligned}

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

30, 60, 90, 120, 150

240

$1$ and $2$

2\times{3}\times{5}

The factor pairs of $30$ are,

\begin{aligned} &1\times{30}\\\\ &2\times{15}\\\\ &3\times{10}\\\\ &5\times{6} \end{aligned}

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 $2.$

1, 2, 3, 5, 6

30\times{5}=150

30, 60, 90, 120, 150

1, 2, 3, 5, 6

\begin{aligned} &30\times{1}=30 \\\\ &30\times{2}=60 \\\\ &30\times{3}=90 \\\\ &30\times{4}=120 \\\\ &30\times{5}=150 \end{aligned}

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

15, 30, 45, 60

60

3.75

15

15\times{4}=60

4, 8, 12, 16, 20, 24

12, 24

1

24, 36

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 $3?$

(d) Which number has exactly $5$ factors?

(5 marks)

Show answer

(a) $4, 12, 16, 24$

(1)

(b) $3, 6,12, 24$

(1)

(2)

(d) $16$

(1)

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?

(3 marks)

Show answer

Multiples of $30$ and $40$ listed.

(1)

$120$ minutes $= 2$ hours

(1)

$9$ : $40am$

(1)

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 $8cm?$

(4 marks)

Show answer

(a)

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

(1)

$4$ rectangles

(1)

(b)

\cfrac{8-[2, 4, 6]}{2}

(1)

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

(1)

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

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