Lesson 5, Topic 1
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# Lowest Common Multiple

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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, 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, 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 = 21 or 2 = 2 x 1

3 = 31 or 3 = 3 x 1

4 = 2 x 2 = 22 or 4 = 2 x 2

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

The LCM = 22 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 factor 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 = 23

Any multiple of 8 must contain 2 x 2 x 2

9 = 3 x 3 = 32

Any multiple of 9 must contain 3 x 3

12 = 2 x 2 x 3 = 22 x 3

Any multiple of 12 must contain 2 x 2 x 3

The lowest product containing all three is 23 x 32

:- The LCM of 8, 9 and 12 is 23 x 32 = 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 = 22 x 3

2 x 3 x 3 x 5 = 2 x 32 x 5

2 x 2 x 5 = 22 x 5

Select each factor with the highest power.

i.e. 22 x 32 x 5

= 4 x  9  x 5 = 180

Solution 2

2 x 3 x 3 = 21 x 32

2 x 2 x 2 x 3 = 23 x 31

2 x 2 x 3 x 5 = 22 x 3 x

Select each factor with the highest power.

i.e. 23 x 32 x 5

= 8 x  9  x 5

= 360 error: