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) and (a-b)
(a + b)² = a² + 2ab + b²
(a - b)² = a² - 2ab +b²
(a + b)³ = a³ + 3a²b + 3ab² + b³
= a³+b³+3ab(a+b)
(a - b)³ = a³ - 3a²b + 3ab² - b³
= a³-b³-3ab(a-b)
Find the H.C.F and L.C.M of x⁴+x² y² +y⁴ ,
x³–y³ , x³ +x²y +xy²
first expression = x⁴+x² y² +y⁴
= (x² + y² )² – 2x²y² + x²y²
= (x² + y² )² - x²y²
=( x +xy +y ) ( x -xy +y )
second expression = x³–y³ = (x-y) ( x²+xy +y²)
3 expression = x³+x²y+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+a²b+a³
first expression = a⁴+a²b²
=a²(a²+b²)
second expression = a³+b³
= (a+b)(a²–ab+b²)
third expression = ab²+a²b+a³
=a(b²+ab+a²)
H.C.F. = 1
L.C.M = a(a+b)(a+b)(a²–ab+b²)(b²+ab+a²)
Find the H.C.F and L.C.M of the following
expressions
first expression = a²+2ab+b²
= (a+b)(a+b)
second expression = b² - a² +2bc +c²
= b² +2bc +c² - a²
= (b+c)² - a²
= (b + c + a ) (b + c - a )
3 expression = - b² + a² +2ca +c²
= (a+c)² - b²
= (a+c-b) (a+c+ b)
H.C.F = (a+b+c)
L.C.M = (a+b)(a+b)(b + c + a )(b + c - a )(a
+c-b)