# Least Common Multiple

The smallest positive number that is a multiple of two or more numbers.

### Least Common Multiple of 3 and 5:

List the Multiples of each number,

The multiples of 3 are 3, 6, 9, 12, 15, 18, ... etc
The multiples of 5 are 5, 10, 15, 20, 25, ... etc

Find the first Common (same) value:

The Least Common Multiple of 3 and 5 is 15

(15 is a multiple of both 3 and 5, and is the smallest number like that.)

## So ... what is a "Multiple" ?

We get a multiple of a number when we multiply it by another number. Such as multiplying by 1, 2, 3, 4, 5, etc, but not zero. Just like the multiplication table.

Here are some examples:

 The multiples of 4 are: 4,8,12,16,20,24,28,32,36,40,44,... The multiples of 5 are: 5,10,15,20,25,30,35,40,45,50,...

## What is a "Common Multiple" ?

Say we have listed the first few multiples of 4 and 5: the common multiples are those that are found in both lists:

 The multiples of 4 are: 4,8,12,16,20,24,28,32,36,40,44,... The multiples of 5 are: 5,10,15,20,25,30,35,40,45,50,...

Notice that 20 and 40 appear in both lists?
So, the common multiples of 4 and 5 are: 20, 40, (and 60, 80, etc ..., too)

## What is the "Least Common Multiple" ?

It is simply the smallest of the common multiples.

In our previous example, the smallest of the common multiples is 20 ...

... so the Least Common Multiple of 4 and 5 is 20.

## Finding the Least Common Multiple

List the multiples of the numbers until we get our first match.

### Example: Find the least common multiple of 4 and 10:

The multiples of 4 are: 4, 8, 12, 16, 20, ...
and the multiples of 10 are: 10, 20, ...

Aha! there is a match at 20. It looks like this:

So the least common multiple of 4 and 10 is 20

### Example: Find the least common multiple of 6 and 15:

The multiples of 6 are: 6, 12, 18, 24, 30, ...
and the multiples of 15 are: 15, 30, ...

There is a match at 30

So the least common multiple of 6 and 15 is 30

## More than 2 Numbers

We can also find the least common multiple of three (or more) numbers.

### Example: Find the least common multiple of 4, 6, and 8

Multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, ...
Multiples of 6 are: 6, 12, 18, 24, 30, 36, ...
Multiples of 8 are: 8, 16, 24, 32, 40, ....

So 24 is the least common multiple (I can't find a smaller one!)

Hint: We can have smaller lists for the bigger numbers.

## Least Common Multiple Tool

There is another method: the Least Common Multiple Tool does it automatically. (Yes, we waited until the end to tell you!)