LCM & HCF and Perfect Squares

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In today’s Mathematics class, We will be discussing LCM, HCF and Perfect Squares. We hope you enjoy the class!

 

Highest Common Factors

A highest common factor is the greatest number which will divide exactly into two or more numbers. For example, 4 is the highest common factor (HCF) of 20 & 24.

i.e. 20 = 1, 2, (4), 5, 10, 20

24= 1, 2, 3, (4), 6, 8, 12, 24

 

Example:

Find the H.C.F of 24 & 78

 

Method 1

Express each number as a product of its prime factors

Workings

2       |   24                    2       |   78

2       |   12                    2       |   36

2       |  6                      2        |  18

3       |   3                      3       |   9

3       |  3

 

24=2³x3

78 =(2³ x 3) x 3

 

The H.C.F. is the product of the common prime factors.

HCF=2³x3

=8×3=24

 

Method II

24=2x2x2x3

78=2x2x2x3x3

Common factor=2x2x2x3

HCF=24

 

LCM: Lowest Common Multiple

Multiples of 2 are =2,4,6,8,10,12,14,16,18,20,22,24…

Multiples of 5 are 5, 10, 15, 20, 25, 30, 35, 40

Notice that 10 is the lowest number which is a multiple of 2 & 5.10 is the lowest common multiple of 2& 5

Find the LCM of 20, 32, and 40

 

Method 1

Express each number as a product of its prime factors

20=2²x5

32=2⁵

40=2²x2x5

The prime factors of 20, 32 and 40 are 2 & 5. The highest power of each prime factor must be in the LCM

These are2⁵ and 5

Thus LCM =2⁵ x5

=160

 

Method II

2      |    20        32        40

2      |    10        16        20

2      |    5          8          10

4      |    5          4          5

5      |    5          1          5

1      |    1          1

 

LCM =2 x 2 x 2 x 4 x 5 = 160

 

 

 

Classwork

Find the HCF of:

(1)  28 and 42

(2)  504 and 588

(3)  Find the LCM of 84 & 210

 

READING ASSIGNMENT

New General Mathematics, UBE Edition, chapter 1, pages 20-21

Essential Mathematics by A J S Oluwasanmi, Chapter 1, Pages 1-4

 

PERFECT SQUARES

A perfect square is a whole number whose square root is also a whole number .It is always possible to express a perfect square in factors with even indices.

9 = 3×3

25= 5×5

225 = 15×15

= 3x5x3x5

= 3² x 5²

 

9216 =96 ²

=3² x 32 ²

=3² x 4² X 8² 

=3²x2⁴ x2⁶ 

=3² x2 ¹⁰

 

Workings

2        |        9216

2        |        4608

2        |        2304

2        |        1152

2        |        576

2        |        288

2        |        144

2        |        72

2        |        36

2        |        18

3        |        9

3        |        3

 

9216= 3²x2¹⁰

 

Example

Find the smallest number by which the following must be multiplied so that their products are perfect square

1. 540
2. 252

Solution

2       |         540

2       |         270

3       |         135

3       |            45

3       |             15                54=2² x 3³x 5

5       |               5

1       |

The index of 2 even. The index of 3 and 5 are odd. One more 3 and one more 5 will make all the indices even. The product will then be a perfect square. The number required is 3×5 =15

 

(2)

2    |         252

2    |        126

3    |         63

3    |         21

7    |         7

1    |

 

252= 2²x3²x7

Index of 7 is odd, one more “7” will make it even.

Indices i.e. 2²x 3²x 7²

Therefore 7 is the smallest numbers required

 

READING ASSIGNMENT

New General Mathematics, UBE Edition, chapter 1, pages 20-21

Essential Mathematics by A J S Oluwasanmi, Chapter 1, pages 1-4

 

WEEKEND ASSIGNMENT

1. The lowest common multiple of 4, 6 and 8 is (a) 24 (b) 48 (c) 12 (d) 40
2. Find the smallest number by which 72 must be multiplied so that its products will give a perfect square (a) 3 (b) 2 (c) 1 (d) 5
3. The lowest common multiple of 4, 6 and 8 is (a) 24 (b) 48 (c) 12 (d) 40
4. The H.C.F. of 8, 24 and 36 is ___ (a) 6 (b) 4 (c) 18 (d) 20
5. The L.C.M. of 12, 16 and 24 is ___ (a) 96 (b) 48 (c) 108 (d) 24

 

THEORY 

1. Find the smallest number by which 162 must be multiplied so that its product will give a perfect square.
2. Find the HCF and L.C.M. of the following figures

• • • 30 & 42
    • 64 & 210

 

 

We have come to the end of this class. We do hope you enjoyed the class?

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In our next class, we will be talking about Fractions as Ratios, Decimals and Percentages. We are very much eager to meet you there.

 

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