If a and b are distinct integers, prove that a – b is a factor of aⁿ – bⁿ, whenever n is a positive integer. [Hint write aⁿ = (a – b + b)ⁿ and expand]

Question

If a and b are distinct integers, prove that a – b is a factor of a n – b n , whenever n is a positive integer. [Hint write a n = (a – b + b) n and expand]

Philoid · Accepted Answer

We can write aⁿ as

aⁿ = (a-b+b)ⁿ

We know that-

putting a = b & b = a-b, we get-

where

Hence (a-b) is a factor of (aⁿ-bⁿ).

If a and b are distinct integers, prove that a – b is a factor of an – bn, whenever n is a positive integer. [Hint write an = (a – b + b)n and expand]

If a and b are distinct integers, prove that a – b is a factor of aⁿ – bⁿ, whenever n is a positive integer. [Hint write aⁿ = (a – b + b)ⁿ and expand]