# Least Common Multiple

*The smallest positive 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,

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!)