Least Common Multiple

The smallest (non-zero) number that is a multiple of two or more numbers.

Let's start with an Example ... 

Least Common Multiple of 3 and 5:

• List the Multiples of each number,

• Find the first Common (same) value

The Least Common Multiple of 3 and 5 is 15
15 is a common multiple of 3 and 5, and is the smallest (least) common multiple

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: You can have smaller lists for the bigger numbers.

Least Common Multiple Tool

There is another method: the Least Common Multiple Tool does it automatically.