Question

(59)^8 - (57)^8 is exactly divisible by which of the following.
a.116 b.115 c.114 d.113​

Answer 1

Answer:

( a ) 116

Step-by-step explanation:

To find---> ( 59 )⁸ - ( 57 )⁸ is exactly

divisible by which number from given number.

Solution--->

( 59 )⁸ - ( 57 )⁸

We have an identity

a² - b² = ( a + b ) ( a - b )

Applying it here

(59)⁸ - (57)⁸ = { (59)⁴}² - { (57)⁴}²

= (59⁴ + 57⁴) (59⁴ - 57⁴)

= (59⁴ + 57⁴) { (59²)² - (57²)² }

Again applying above identity

=( 59⁴ + 57⁴) ( 59² + 57²) ( 59² - 57²)

Again applying above identity

=( 59⁴+57⁴) (59²+57²) (59+57 )(59-57)

=(59⁴ + 57⁴) (59² + 57² ) ( 116 ) ( 2 )

= 2 × 116 × ( 59² + 57² ) ×( 59⁴ + 57⁴ )

Clearly ( 59 )⁸ - ( 57 )⁸ is divisible by 116 because 116 is factor of it.

Additional information--->

1) ( a + b )² = a² + b² + 2a b

2) ( a - b )² = a² + b² - 2a b

3) (a+b +c )²=a²+b²+c²+2ab +2bc +2ca

4) (a + b )³ = a³ + b³ + 3 a b ( a + b )

5) (a - b )³ = a³ - b³ - 3 a b ( a - b )