# Highest Common Factor and Lowest Common Multiple

Highest common Factor and lowest common multiple

The highest common factor is calculated by multiplying all the factors which appear in both lists: So the HCF of 60 and 72 is 2 × 2 × 3 which is 12. The lowest common multiple is calculated by multiplying all the factors which appear in either list: So the LCM of 60 and 72 is 2 × 2 × 2 × 3 × 3 × 5 which is 360.

Some important formulas

Difference of squares

a2 - b2 = (a-b)(a+b)

Difference of Cubes

a3 - b= (a - b)(a2+ ab + b2)

Sum of Cubes

a3 + b3 = (a + b)(a2 - ab + b2)

Formula for (a+b)2 and (a-b)2

(a + b)2 = a2 + 2ab + b2

(a - b)2 = a2 - 2ab +b2

(a + b)3 = a3 + 3a2b + 3ab2 + b3

(a - b)3 = a3 - 3a2b + 3ab2 - b3

Find the H.C.F and L.C.M of x4 +x2 y2 +y4 , x3 –y3 , x3 +x2 y  +xy2

1st expression   = x4 +x2 y2 +y4

= (x2 + y2)2 – 2x2y2 + x2y2

= (x2 + y2)2  - x2y2

=( x2  +xy  +y2 ) ( x2  -xy  +y2 )

2nd expression = x3 –y3 = (x-y) ( x2  +xy  +y2 )

3rd expression = x3 +x2 y +xy2 = x( x2  +xy  +y2 )

H.C.F = ( x2  +xy  +y2 )

L.C.M = x ( x2  +xy  +y2 ) ( x2  -xy  +y2 ) (x-y)

Find the H.C.F and L.C.M of a4 +a2b2 , a3 +b3 , ab2 +a 2b +a3

1st expression = a4 +a2b2

=a2(a2  + b2 )

2nd expression = a3 +b3

= (a+b)(a2 –ab + b2 )

3rd expression = ab2 +a 2b +a3

= a(b2 +ab +a2 )

HCF = 1

L.C.M =  a2 (a+b) (a2  + b2 ) (a2 –ab + b2 ) (b2 +ab +a2 )

Find the H.C.F and L.C.M of the following expressions

1st expression = a2 +2ab + b2

= (a+b)(a+b)

2nd expression = b2 -  a2 +2bc +c2

=  b2 +2bc +c2 -  a2

= (b+c)2 - a2

= (b + c + a ) (b + c  -   a )

3rd expression = - b2 +   a2 +2ca +c2

=  (a+c)2  - b2

= (a+c-b)  (a+c+ b)

H.C.F  = 1

L.C.M  =  (a+b)(a+b)(b + c + a )(b + c  -   a )(a+c-b)

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