Least Common Multiple

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

Least Common Multiple is made up of the words Least, Common and Multiple:

What is a "Multiple" ?

The multiples of a number are what you get when you multiply it by other numbers (such as if you multiply it by 1,2,3,4,5, etc). Just like the multiplication table.

Here are some examples:

The multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, etc ...

The multiples of 12 are: 12, 24, 36, 48, 60, 72, etc...

What is a "Common Multiple" ?

When you list the multiples of two (or more) numbers, and find the same value in both lists, then that is acommon multiple of those numbers.

For example, when you write down the 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

It is a really easy thing to do. Just start listing the multiples of the numbers until you get a match.

Example: Find the least common multiple for 3 and 5:

The multiples of 3 are 3, 6, 9, 12, 15, ..., 
and the multiples of 5 are 5, 10, 15, 20, ..., like this:

As you can see on this number line, the first time the multiples match up is 15. Answer: 15

More than 2 Numbers

You can also find the least common multiple of 3 (or more) numbers.

Example: Find the least common multiple for 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 !)