Class 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 - 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)

                                                                                                                                                                                                                      


Go Top