Title: LESSON: Least Common Multiple
Course: Mathematics Essentials Q1 > Section: Week 7

Least Common Multiple

INTRODUCTION

Least Common Multiple

READ: Find the Common Multiples of Different Numbers

Find the Common Multiples of Different Numbers

In mathematics, you have been working with multiples for a long time. One of the first things that you probably learned was how to count by twos or threes. Counting by twos and threes is counting by multiples. When you were small, you didn’t call it “counting by multiples,” but that is exactly what you were doing.

What is a multiple? A multiple is the product of a quantity and a whole number.

What does that mean exactly? It means that when you take a number like 3 that becomes the quantity. Then you multiply that quantity times different whole numbers.

3 2 6, 3 3 9, 3 4 12, 3 5 15, 3 6 18

Listing out these products is the same as listing out multiples.

3, 6, 9, 12, 15, 18.....

You can see that this is also the same as counting by threes.

The dots at the end mean that these multiples can go on and on and on. Each numbers has an infinite number of multiples.

Example

List five the multiples for 4.

To do this, we can think of taking the quantity 4 and multiplying it by 2, 3, 4, 5, 6.....

4 2 8, 4 3 12, 4 4 16, 4 5 20, 4 6 24

Our answer is 8, 12, 16, 20, 24.... Notice that we could keep on listing multiples of 4 forever.

What is a common multiple? A common multiple is a multiple that two or more numbers have in common.

Example

What are the common multiples of 3 and 4?

To start to find the common multiples, we first need to write out the multiples for 3 and 4. To find the most common multiples that we can, we can list out multiples through multiplying by 12.

3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36

4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48

The common multiples of 3 and 4 are 12, 24, 36.

EXAMPLE 1

EXAMPLE 2

READ: Find the LCM Using Lists

Find the Least Common Multiple of Given Numbers Using Lists

We can also find the least common multiple of a pair of numbers.

What is the least common multiple? The least common multiple (LCM) is just what it sounds like, the smallest multiple that two numbers have in common.

Let’s look back at the common multiples for 3 and 4.

3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36

4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48

Here we know that the common multiples are 12, 24 and 36.

The LCM of these two numbers is 12. It is the smallest number that they both have in common. We used lists of multiples for 3 and 4 to find the common multiples and then the least common multiple.

READ: Find the LCM using Prime Factorization

Find the Least Common Multiple of Given Numbers Using Prime Factorization

Remember back to factoring numbers? We worked on using factor trees to factor numbers or to break down numbers into their primes. Take a look at this one.

We used a factor tree in this example to factor twelve down to the prime factors of 2 squared times 3.

We can also use prime factorization when looking for the least common multiple.

How can we use prime factorization to find the LCM?

If we wanted to find the LCM of two numbers without listing out all of the multiples, we could do it by using prime factorization.

Example

What is the LCM of 9 and 12?

First, we factor both numbers to their primes.

Next, we identify any shared primes. With 9 and 12, 3 is a shared prime number.

Then, we take the shared prime and multiply it with all of the other prime factors.

3 3 2 2

The red 3 is the shared prime factor.

The blue numbers are the other prime factors.

Our answer is 36. The LCM of 9 and 12 is 36.

EXAMPLE 3

CHECK Yourself! Least Common Multiple