problem solving with lcm

Least Common Multiple

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

Let's start with an Example ... 

Example: 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:

What is a "Common Multiple" ?

When we list the multiples of (for example) 4 and 5 , the common multiples are those that are found in both lists:

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, ...

The first match is 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 (we 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!)