# What are Multiples?

Multiples are sets of numbers that correspond to a base number that has been multiplied by other numbers. For example, the multiples of 3 include 3 (3×1), 6 (3×2), 9 (3×3), 12 (3×4), 15 (3×5), etc.; the multiples of 5 include, 5, 10, 15, 20, 25, etc.

## Multiples

Hello and welcome to this video about multiples! In this video, we’ll take a look at

- What multiples are
- Common multiples
- and Least Common Multiples

So first thing’s first – what is a multiple? Well, a multiple is any **number** obtained by multiplying other numbers together. Multiples are most commonly discussed in the context of integers. Let’s begin with a simple multiplication table:

3×0=0 | The numbers 0, 3, 6, 9, and 12 are all multiples of 3. In order to obtain them, 3 is being multiplied by another number. Any number has an infinite amount of multiples. |

3×1=3 | |

3×2=6 | |

3×3=9 | |

3×4=12 |

Some numbers are multiples of many different numbers. Take 12 for example:

1×12=12 | 12 is a multiple of 1, 2, 3, 4, 6 and 12. These numbers can be multiplied by other numbers to obtain 12. As a reminder, 1, 2, 3, 4, 6 and 12 would be called factors of 12. |

2×6=12 | |

3×4=12 |

Other numbers are only multiples of 2 numbers, for example, 7:

1×7=7 | 7 is a multiple of 1, and 7 only (1 and 7 are the only factors of 7). Because of this, 7 is an example of a prime number. |

Now that we know what multiples are, here are a couple of truths to remember about them:

- First, 0 is a multiple of all numbers. For any number
*a*: a∙0=0 - and secondly, every number is a multiple of 1 and itself. For any number
*a*, a∙1=a

When comparing numbers, sometimes they are compared by their **common multiples**, or multiples that the numbers have in common.

Here are some common multiples of 4 and 10:

4×5=20 | 10×2=20 | 20, 40, 60, 80, and 100 are all multiples of both 4 and 10. |

4×10=40 | 10×4=40 | |

4×15=60 | 10×6=60 | |

4×20=80 | 10×8=80 | |

4×25=100 | 10×10=100 |

We can find common multiples of more than two numbers as well. Let’s look at 3, 5, and 9:

3×15=45 | 5×9=45 | 9×5=45 | 45, 90, and 135 are all common multiples of 3, 5 and 9. |

3×30=90 | 5×18=90 | 9×10=90 | |

3×45=135 | 5×27=135 | 9×15=135 |

Sometimes, we need to determine the **Least Common Multiple (LCM)** of a set of numbers, in other words, the common multiple that has the lowest value.

This can be done in one of two ways: listing the common multiples and factoring. Before, we listed the common multiples of two different sets of numbers. The LCM of 4 and 10 is 20, while the LCM of 3, 5 and 9 is 45.

Now let’s find the LCM of 4 and 10 by factoring. Remember, *factors* are numbers that are multiplied together to form a multiple. The LCM of two numbers is the product of the highest power of all *prime* factors of both numbers. Let’s break that down a bit:

4=2×2 | 10=5×2 |

The prime factor of 4 is 2. | The prime factors of 10 are 5 and 2. |

The factor of 2 occurs twice, so it is represented as 2^{2}. So, 4=2^{2}. | |

The unique factors here are 2 and 5 and the highest power of 2 is 2, so the LCM equals 2^{2}×5=20, just as we determined by making a list. |

Now let’s find the LCM of 24, 144, and 48 by factoring:

24=23×3 | 144=12×12=2432 | 48=16×3=24×3 |

The prime factors of 24 are 2 and 3. | The prime factors of 144 are 2 and 3. | The prime factors of 48 are 2 and 3. |

The unique factors here are 2 and 3. The highest power of 2 is 4 and the highest power of 3 is 2. The LCM equals 2432=144. Since 144 is a multiple of both 24 and 48, it is also automatically the LCM of all three numbers. |

I hope that this video has increased your understanding of multiples and how they work with numbers!

See you next time!