# Finding the multiples of a number

## What is a multiple?

It’s helpful to think about multiples and divisibility as two parts of the same idea. We know that ???10??? is “divisible” by ???5??? because when we do the division ???10\div5???, the result ???2??? is a whole number. It’s the fact that the result is a whole number that proves that ???10??? is divisible by ???5???.

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To understand the concept of “multiples,” we need to turn around our thinking about divisibility. So instead of thinking of ???10\div5=2???, let’s think about ???2\times5=10???. This multiplication problem tells us that ???10??? is a “multiple” of both ???2??? and ???5???.

A multiple of a number is what we get when we multiply that number by a whole number. Here’s how we get some of the multiples of ???2???:

???2\times1=2???

???2\times2=4???

???2\times3=6???

???2\times4=8???

???2\times5=10???

So ???2???, ???4???, ???6???, ???8???, and ???10??? are multiples of ???2???. And here’s how we get some of the multiples of ???5???:

???5\times1=5???

???5\times2=10???

???5\times3=15???

???5\times4=20???

???5\times5=25???

So ???5???, ???10???, ???15???, ???20???, and ???25??? are multiples of ???5???.

The multiples of ???2??? that we’ve listed here aren’t the only multiples of ???2???, and the multiples of ???5??? that we’ve listed here aren’t the only multiples of ???5???. They’re just a few of the multiples of those numbers, to give you an idea of what a “multiple” is.

To relate multiples to divisibility, you now know two reasons why ???20??? is a multiple of ???5???:

1. ???20??? is a multiple of ???5??? because ???20??? is divisible by ???5???, since ???20\div5??? gives a whole number answer of ???4???.

2. ???20??? is a multiple of ???5??? because ???5??? multiplied by a certain whole number, ???4???, gives ???5\times4=20???.

## How to determine if one number is a multiple of another

Example

Is ???80??? a multiple of ???10???? If so, give two reasons why.

Yes, ???80??? is a multiple of ???10???, for two reasons:

1. ???80??? is a multiple of ???10??? because ???80??? is divisible by ???10???, since ???80\div10??? is ???8???, a whole number.

2. ???80??? is a multiple of ???10??? because ???10??? multiplied by a certain whole number, ???8???, gives ???10\times8=80???.

It’s helpful to think about multiples and divisibility as two opposite parts of the same idea.

We could think of ???0??? as a multiple of any number, because if we multiply any number by ???0???, we get ???0???. For example, we could think of the multiples of ???2??? as ???0???, ???2???, ???4???, ???6???, ..., which are also known as even numbers. However, there is no number that’s divisible by ???0???, because division by ???0??? is never allowed!