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:
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:
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:
We can use multiplication tables to help us work out factors of a number.
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:
Complete this table of factor pairs:
Factor pairs of $75$75 |
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$\left(\editable{},75\right)$(,75) |
$\left(\editable{},25\right)$(,25) |
$\left(5,\editable{}\right)$(5,) |
Write down all the factors of $10$10.
Write each answer on the same line, separated by commas.
Write the first five multiples of $8$8, separating each with a comma.