H.C.F & L.C.M AND PERFECT SQUARES

Highest Common Factors

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 1:

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

 

Class work

Find the HCF of:

(1)  28 and 42

(2)  504 and 588

(3)  Find the LCM of 84 & 210

 

 

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

1. 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

 

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

 

See also

WHOLE NUMBERS AND DECIMAL NUMBERS

BASIC OPERATION OF INTEGERS

WEIGHT

VOLUME OF CYLINDER

AREA OF RIGHT ANGLED TRIANGLE