Close sidebar
Skip to content

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

See also  SOLVING PROBLEMS BASED ON LAWS OF LOGARITHM

=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 8

=32x24 x2

=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.

See also  FACTORS

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

SUBSCRIBE BELOW FOR A GIVEAWAY

Building & maintaining an elearning portal is very expensive, that is why you see other elearning websites charge fees. Help to keep this learning portal free by telling mum or dad to donate or support us. Thank you so much. Click here to donate

Leave a Reply

Your email address will not be published.

error: Content is protected !!