1.10 Factors and multiples

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

Every number will have at least two factors, 1 and the number itself. Thinking again about the number 12, we know that 12 \div 1 = 12 with no remainder, and 12 \div 12 = 1 with no remainder, so 12 and 1 are both factors of 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 by starting at 1 and trying to find a number that multiplies with it to make 12:

\begin{array}{c} &1 &\times &12 &= &12 \\ &2 &\times &6 &= &12 \\ &3 &\times &4 &= &12 \end{array}

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

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

Watch this video for more examples:

We can use multiplication tables to help us work out factors of a number.

Examples

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

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

\begin{array}{c} &2 &\times &1 &= &2 \\ &2 &\times &2 &= &4 \\ &2 &\times &3 &= &6 \end{array}

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

Here are some other examples:

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

Examples