Khullakitab.comClass 10 Mathematics Note
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 - 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)