Question

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]

Answer 1

In order to prove that (a – b) is a factor of (aⁿ – bⁿ), it has to be proved that

aⁿ – bⁿ = k (a – b), where k is some natural number

It can be written that, a = a – b + b

This shows that (a – b) is a factor of (aⁿ – bⁿ), where n is a positive integer.