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 x 1, 2 x 2, 2 x 3, 2 x 4, 2 x 5, 2 x 6

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

The multiples of 4 are:

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

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

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

For example the common multiple 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

### Example 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 number

**Method 1** Listing Method

The multiple 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 that 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 = 2^{1} or 2 = 2 x 1

3 = **3 ^{1}** or 3 = 3 x 1

4 = 2 x 2 = **2 ^{2}**

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

The highest power of 2 is 2^{2}

The highest power of 3 is 3

Product = The LCM = 2^{2} x 3

= 4 x 3

= 12

### Example 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, …

L.C.M = 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** x 1

5 = **5** x 1

6 = **2** x 3

The LCM = 2 x 3 x 5 = 30

### Example 3

Find the LCM of 8, 9 and 12.

**Solution **

8 = 2 x 2 x 2 = **2 ^{3}**

Any multiple of 8 must contain 2 x 2 x 2

9 = 3 x 3 = **3 ^{2}**

Any multiple of 9 must contain 3 x 3

12 = 2 x 2 x 3 = 2^{2} x 3

Any multiple of 12 must contain 2 x 2 x 3

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

= 2^{3} x 3^{2}

⇒ The LCM of 8, 9 and 12

= 2^{3} x 3^{2} = 8 x 9 = 72

### Example 4

Find the LCM of the following. Leave the answers as prime factors. **Question 1**

2 x 2 x 3

2 x 3 x 3 x 5

2 x 2 x 5 **Question 2**

2 x 3 x 3

2 x 2 x 2 x 3

2 x 2 x 3 x 5

**Solution 1**

2 x 2 x 3 = **2 ^{2}** x 3

2 x 3 x 3 x 5 = 2 x **3 ^{2}** x

**5**

2 x 2 x 5 = **2 ^{2}** x

**5**

Select each factor with the highest power.

i.e. 2^{2} x 3^{2} x 5

= 4 x 9 x 5 = 180

**Solution 2**

2 x 3 x 3 = 2^{1} x **3 ^{2}**

2 x 2 x 2 x 3 = **2 ^{3}** x 3

^{1}

2 x 2 x 3 x 5 = 2^{2} x 3 x **5 **

Select each factor with the highest power.

i.e. 2^{3} x 3^{2} x 5

= 8 x 9 x 5

= 360

Easy and straightforward