Lowest Common Multiple

Topic Content:

The multiples of a number are all the numbers that are products of the number and any other integer.

For example:

The multiples of 2 are:

2 × 1, 2 × 2, 2 × 3, 2 × 4, 2 × 5, 2 × 6

which are 2, 4, 6, 8, 10, 12, 14, 16, and so on.

The multiples of 4 are:

4 × 1, 4 × 2, 4 × 3, 4 × 4, 4 × 5, 4 × 6 …..

which are  4, 8, 12, 16, 20, 24, …. etc

What is the Lowest Common Multiple?

The Lowest Common Multiple (LCM) of two or more numbers is the lowest multiple they have in common.

For example, the common multiples of 2 and 3.

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22

3, 6, 9, 12, 15, 18, 21, 24

Common multiples are 6, 12, 18 …

The smallest or lowest of these multiples is 6. 

This means 6 is the LCM 

Worked Example 5.2.1:

Find the first three common multiples of 2, 3 and 4. What is their LCM? 

Solution 

There are two common methods for solving LCM. 

1. Listing method 
2. Product of their prime factors or Prime factor method

Method 1 – Listing Method:

The multiples of 2, 3 and 4 

2 = 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36

3 = 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45

4 = 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52

⇒ The first three common multiples of 2, 3, and 4 are 12, 24 and 36 

The LCM is 12

(12 is the smallest/lowest multiple common to 2, 3 and 4, which is why 12 is the LCM, Lowest Common Multiple). 

Method 2:

You can also find the LCM of a set of numbers by using the products of their prime numbers. 

2 = 2¹ or 2 = 2 × 1 

3 = 3¹ or 3 = 3 × 1 

4 = 2 × 2 = 2²

By selecting each prime factor with the “highest power”. 

The highest power of 2 is 2²
The highest power of 3 is 3

Product = The LCM = 2² × 3 

= 4 × 3 

= 12

Worked Example 5.2.2:

Find the LCM of 3, 5 and 6 using the above two methods. 

Solution 

Method 1 :

The multiples of 3, 5 and 6 are 

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

5 = 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, ….

6 = 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, … 

LCM = 30. 

It means 3, 5 and 6 can divide 30 without any remainder. 

Using the second Method:

The prime factors of 3, 5 and 6 are 

3 =  3 × 1 

5 =  5 × 1 

6 =  2 × 3 

The LCM = 2 × 3 × 5 = 30  

Worked Example 5.2.3:

Find the LCM of 8, 9 and 12. 

Solution 

8 = 2 × 2 × 2 = 2³

Any multiple of 8 must contain 2 × 2 × 2 

9 = 3 × 3 = 3²

Any multiple of 9 must contain 3 × 3 

12 = 2 × 2 × 3 = 2² × 3

Any multiple of 12 must contain 2 × 2 × 3 

By selecting each prime factor with the “highest power”. 

= 2³ × 3²

⇒ The LCM of 8, 9 and 12:

= 2³ × 3²

= 8 × 9

= 72 

Worked Example 5.2.4:

Find the LCM of the following. Leave the answers as prime factors. 

Question 1

2 × 2 × 3
2 × 3 × 3 × 5
2 × 2 × 5 

Question 2

2 × 3 × 3
2 × 2 × 2 × 3
2 × 2 × 3 × 5

Solution 1

2 × 2 × 3 = 2² × 3 

2 × 3 × 3 × 5 = 2 × 3² × 5 

2 × 2 × 5 = 2² × 5 

Select each factor with the highest power. 

i.e. 2² × 3² × 5 

= 4 ×  9  × 5 = 180 

Solution 2

2 × 3 × 3 = 2¹ × 3²

2 × 2 × 2 × 3 = 2³ × 3¹ 

2 × 2 × 3 × 5 = 2² × 3 × 5 

Select each factor with the highest power. 

i.e. 2³ × 3² × 5 

= 8 × 9 × 5 

= 360