# Definitions - Factors and Multiples

In Math . . .

If a is a number and b divides a evenly then b is a factor of a and a is a multiple of b.

In English . . .

There are really two parts of this definition: factors and multiples. If one number goes into another number evenly then the second number is a factor of the first. For example, because 3 divides 6 evenly, i.e. without a remainder, 3 is a factor of 6. Similar because 2 divides 6 evenly, 2 is also a factor of 6.

Multiples are the opposite of factors. If I give you a number and you multiply it by an integer then the new number is a multiple of the first. For example, because 5 · 6 = 30, 30 is a multiple of 5. (It's also a multiple of 6.)

## Quick Tip - Which One Did You Find?

I often see students who are asked to find a multiple accidentally finding a factor instead. Here's a quick rule of thumb you can use to make sure you went in the right direction: Multiples are always bigger than the original number and factors are always smaller.

So if you're given a number, how can you find a factor or a multiple of it? There are two simple procedures that you can use.

1. Finding a Factor: To find a factor of a number, start dividing it by smaller numbers. When you find one that goes into it evenly, that number is a factor of the original one.
2. Finding a Multiple: To find a multiple of a number, take the original number and multiply it by an integer. The new number is a multiple of your original one.

# Videos

### Lectures

We're going to start our conversation about the properties of integers by looking at factors and multiples. We'll talk about their definitions then look at some specific examples and the relationship between the two. (lecture slides)
Now that we've seen what factors are, we're going to look at how to list all the factors of a number. We'll look at a couple of examples then compare the results. (lecture slides)

# Factor Examples

List all the factors of 18.

To answer a question like this, we need to write down all the combinations of numbers that, when we multiply them together, equal 18.

 18 x 1 = 18 9 x 2 = 18 6 x 3 = 18 6 x 3 = 18 2 x 9 = 18 1 x 18 = 18

This tells me that the factors of 18 are 1, 2, 3, 6, 9 and 18.

Is 18 a factor of 112?

This is the same as asking, does 18 go into 112 evenly? Since 112 / 18 = 6.222... which isn't an integer, we can conclude that 18 isn't a factor of 112.

List two numbers that 9 is a factor of.

Because 9 goes into both 18 and 27 evenly, it must be a factor of 18 and 27.

# Multiple Examples

Is 110 a multiple of 10?

To test this, you would take 110 and divide it by 10. If it comes out to an even number then the answer is yes. If you do the arithmetic, you'll see that 110 / 10 = 11. Since 11 is an integer, we can conclude that 110 is a multiple of 10.

List two multiples of 15.

To find a muliple of a number, you need to take the number and multiply it by any integer. For example, 15 x 2 = 30 and 15 x 3 = 45 so 30 and 45 are both multiples of 15.