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.
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.|
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.|
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.|
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.|
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:
|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 22. So, 4=22.|
|The unique factors here are 2 and 5 and the highest power of 2 is 2, so the LCM equals 22×5=20, just as we determined by making a list.|
Now let’s find the LCM of 24, 144, and 48 by factoring:
|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!