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.
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:
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.
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 times 0, equals 0.
And secondly, every number is a multiple of 1 and itself.
For any number a, a times 1, equals 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.
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.
45, 90, and 135 are all common multiples of 3, 5 and 9.
Sometimes, we need to determine the Least Common Multiple (or 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 equals 2 times 2.
The prime factor of 4 is 2.
The factor of 2 occurs twice, so it is represented as 2 squared.
So, 4 equals 2 squared.
10 equals 5 times 2.
The prime factors of 10 are 5 and 2.
The unique factors here are 2 and 5 and the highest power of 2 is 2, so the LCM equals
2 squared times 5 which is equal to 20, just as we determined by making a list.
Now let’s find the LCM of 24, 144, and 48 by factoring.
24 equals, 2 cubed times 3
So the prime factors of 24 are 2 and 3.
144 is equal to 12 times 12, which is equal to 2^4 times 3 squared.
Which means that the prime factors are also 2 and 3.
48 is equal to 16 times 3, which is equal to 2^4 times 3.
So once again, 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.
So the LCM equals 2^4 times 3 squared, which is equal to 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!
Thanks for watching, and happy studying!