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# Lowest Common Multiple (LCM) and Highest Common Factor (HCF) of Whole Numbers Continues.

Welcome to class!

In today’s class, we will be talking more about the lowest common multiple (LCM) and the highest common factor (HCF) of whole numbers. Enjoy the class!

### Lowest Common Multiple (LCM) and Highest Common Factor (HCF) of Whole Numbers

• #### The lowest common multiples (LCM) of two whole numbers

From the list of multiples of two whole numbers, we can find the list of their common multiples. We pick from the list, the lowest or least common multiples which is usually called L.C.M

Let us consider the following examples.

The multiples of 12 are:

12, 24, 36, 48, 60, 72, 84, 96, ….

The multiples of 18 are:

18, 36, 54, 72, 90, 108, …

The common multiples of 12 and 18 are 36, 72, 108, …

The L.C.M of 12 and 18 is 36.

The result may be obtained by expressing 12 and 18 as products of prime factors:

12 = 2 X 2 x 3

18 = 2 x 2 x 3

The L.C.M. of 12 and 118 = 2 x 2 x 3 x 3 = 30.

Note: that each prime factor in 12 and 18 is taken for their L.C.M. and from where it occurs most; the two’s in rather than the only one in 18 and the two 3’s in 18 rather than the only one in 12.

##### Example 1

Find the L.C.M. of 42 and 96.

42 = 2 x 3 x 7

96 = 2 x 2 x 2 x 2 x 2 x 3

The L.C.M. of 42 and 96 is 2 x 2 x 2 x 2 x 2 x 3 x 7 = 672

Note that 3 occurs once for both 42 and 96 and so we take 3 once for the L.C.M. 7 occurs in 42 and not in 96, so we take 7 once for the L.C.M.

###### Activity 1

Find the L.C.M. of (i) 16 and 30, (ii) 18 and 12, by the method of listing common multiples and by the method of prime factors. Which method do you prefer? Give reasons.

##### Example 2

Find the L.C.M of 6, 5 and 4.

6 = 2 x 3

5 = 5 x 1

4 = 2 x 2

: · L.C.M = 2 x 2 x 3 x 5 = 60

##### Example 3

Find the H.C.F of 13 and 39.

13 = 1 x 13

39 = 3 x 13

H.C.F = 13

##### Example 4

Find the L.C.M of 2² X 3² X 5, 2 x 3² x 5² and 2³ x 3 x 5³

2² x 3² x 5 = 2 x 2 x 3 x 3 x 5

2 x 3² x 5² = 2 x 3 x 3 x 5 x 5

2³ x 3 x 5³ = 2 x 2 x 2 x 3 x 5 x 5 x 5

L.C.M = 2 x 2 x 2 x 3 x 3 x 5 x 5 x 5

= 2³ x 3² x 5³

= 9000

##### Example 5

What is the H.C.F of 21, 45 and 105?

21 = 3 x 7

45 = 5 x 3 x 3

105 = 3 x 5 x 7

: · H.C.F = 3

In our next class, we will be talking about Fractions.  We hope you enjoyed the class.

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