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 - b3 = (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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Notes