1.11 Factors and multiples

A number is a factor if it divides another number with no remainder. For example, $4$4 is a factor of $12$12, because $12\div4=3$12÷​4=3 with no remainder.

Every number will have at least two factors, $1$1 and the number itself. Thinking again about the number $12$12, we know that $12\div1=12$12÷​1=12 with no remainder, and $12\div12=1$12÷​12=1 with no remainder, so $12$12 and $1$1 are both factors of $12$12.

If we're finding all the factors of a number we are trying to find all the numbers we can multiply together to make that specific number.

Let's find the factors of $12$12 by starting at $1$1 and trying to find a number that multiplies with it to make $12$12:

$1$1	$\times$×	$12$12	$=$=	$12$12
$2$2	$\times$×	$6$6	$=$=	$12$12
$3$3	$\times$×	$4$4	$=$=	$12$12

Remember, $4\times3=3\times4$4×3=3×4, and both of these numbers already appear, so we can move to the next number up. $5$5 does not divide evenly into $12$12, and we already know $6$6 is a factor because $6\times2=2\times6$6×2=2×6. Once you have checked all numbers up to half of the number (in this case $6$6 is half of $12$12) you can stop.

So, the factors of $12$12 are all the numbers on the left hand side: $1$1, $2$2, $3$3, $4$4, $6$6 and $12$12. 

Watch this video for some more examples:

Factor pairs

Notice that in the above example, each factor had a corresponding factor it multiplied with to give the 'target' number. These two numbers are known as a factor pair.

We can construct any number by using its factor pairs. Every number will have at least one factor pair, $1$1 and itself. 

When we're completing factor expressions we need to think of a second number we can use to complete an expression. We can do this by counting up or dividing. Let's look at an example:

Worked example

Complete the factor pair below:

$\editable{}$$\times$×$7=21$7=21

Think: Here we need to complete the factor expression by finding what goes in the $\editable{}$.

So far we can see two numbers $7$7 and $21$21. We can use these two numbers to work out the missing number - the number that makes a factor pair with $7$7 to make $21$21. 
We can either find the result of $21\div7$21÷​7 or we can count by $7$7's.

Do: If we count by $7$7's we get: $7$7, $14$14, $21$21...

On a number line it would look like this:

We can now see that we'll have to count by $7$7 three times before we get to $21$21, so $3$3 groups of $7$7 make $21$21. 

So, the number $3$3 completes the factor pair:

$3\times7=21$3×7=21

Reflect: If we know one number in a factor pair, we can find the other number in the pair by dividing the original number by the known number of the pair.

Watch this video for more examples:

 

Multiples

A number is a multiple of another if it can be made by multiplication. For example, $15$15 is a multiple of $5$5, because $15=3\times5$15=3×5.

To find multiples of a number we start by multiplying by $1$1, then by $2$2, then by $3$3, and so on. To find multiples of $2$2 we work our way through starting at

$2\times1=2$2×1=2
$2\times2=4$2×2=4
$2\times3=6$2×3=6

... and can keep going forever! Our work so far tells us that the first three multiples of $2$2 are $2$2, $4$4, and $6$6.

Here are some other examples:

• The first five multiples of $4$4 are $4$4, $8$8, $12$12, $16$16 and $20$20
• The first six multiples of $3$3 are $3$3, $6$6, $9$9, $12$12, $15$15, and $18$18
• The first five multiples of $7$7 are $7$7, $14$14, $21$21, $28$28, and $35$35

Examples

Question 1

Question 2

Question 3