Math Core

You have 12 cupcakes to set out on a tray in equal rows. You could make 2 rows of 6, or 3 rows of 4, or even 1 long row of 12. The numbers that fit are called factors of 12. In this lesson you will learn to find every factor of a number and to list its multiples.

Factor pairs

When two whole numbers multiply to make a number, they are a factor pair for that number. Here are 12 squares arranged as 3 rows of 4:

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Because $3 \times 4 = 12$ , the numbers 3 and 4 are a factor pair of 12. The other rectangles you can make with 12 squares give the other pairs:

Rows	Columns	Equation
1	12	$1 \times 12 = 12$
2	6	$2 \times 6 = 12$
3	4	$3 \times 4 = 12$

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

Definition

Factor

A factor of a number is a whole number that divides it with no remainder. For example, 4 is a factor of 12 because $12 \div 4 = 3$ with nothing left over. 5 is not a factor of 12, because $12 \div 5$ is 2 R2.

Finding all the factors

To find every factor of a number, test 1, 2, 3, and so on, in order. Each time a number divides evenly, write down it and its partner. Stop when the numbers start to repeat.

Here is 24:

$1 \times 24$ ✔
$2 \times 12$ ✔
$3 \times 8$ ✔
$4 \times 6$ ✔
5 does not work ( $24 \div 5$ has a remainder).
The next number, 6, is already in our list. The pairs have met in the middle, so we are done.

The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24.

Tip

Every number has 1 and itself as factors. Write those first, then fill in the middle. Remember the divisibility clues you know: every even number has 2 as a factor, and a number ending in 0 or 5 has 5 as a factor.

Multiples

A multiple of a number is what you get when you multiply it by a whole number. You find multiples by skip counting.

The first multiples of 6 are 6, 12, 18, 24, 30, 36, … They go on forever.

Skip counting by 6 lands on the multiples of 6.

Factors and multiples go together

If $3 \times 8 = 24$ , you can say it two ways:

3 and 8 are factors of 24.
24 is a multiple of 3 and a multiple of 8.

A number has only a few factors, but it has endless multiples.

Worked example: All the factors of 18

List all the factors of 18.

Test in order: $1 \times 18$ , $2 \times 9$ , $3 \times 6$ . Then 4 and 5 do not divide 18 evenly. The next number, 6, is already listed, so stop.

The factors of 18 are 1, 2, 3, 6, 9, 18.

Worked example: Is it a multiple?

Is 42 a multiple of 7? Is 42 a multiple of 8?

$7 \times 6 = 42$ , so yes, 42 is a multiple of 7.

Skip count by 8: 8, 16, 24, 32, 40, 48. You jump from 40 right past 42 to 48, so no, 42 is not a multiple of 8.

Worked example: Factors of 36

List all the factor pairs of 36.

1 \times 36, \quad 2 \times 18, \quad 3 \times 12, \quad 4 \times 9, \quad 6 \times 6

5 does not work. After $6 \times 6$ the pairs would repeat. Notice that 6 pairs with itself, so we write it only once.

The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36. That is 9 factors.

Common mistake

Don't mix up factors and multiples. Factors of 12 are 12 or smaller (1, 2, 3, 4, 6, 12). Multiples of 12 are 12 or bigger (12, 24, 36, …). The number 24 is a multiple of 12, not a factor of it.

Practice

Practice 1

List all the factors of 8.

Separate answers with commas, e.g. 2, -5

Practice 2

List the first five multiples of 5.

Separate answers with commas, e.g. 2, -5

Practice 3

Which number is not a factor of 30?

Practice 4

List all the factors of 20.

Separate answers with commas, e.g. 2, -5

Practice 5

Which number is a multiple of 9?

Practice 6

How many factors does 40 have?

Type a number.

Practice 7

Mr. Lee has 48 chairs. He wants to set them in equal rows with no chairs left over. List every number of chairs that could go in one row.

Separate answers with commas, e.g. 2, -5

Practice 8

I am a multiple of 4 and a multiple of 9. I am between 30 and 40. What number am I?

Type a number.