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.

Answer 1

When n is an even number then $a^n-b^n$ can be written as $(a^{n/2}-b^{n/2})(a^{n/2}+b^{n/2})$. here if $\frac{n}{2}$ is even then it can be further expanded and finally we will get (a-b)*(some thing).

now, here if n or $\frac{n}{2}$ is odd then the expansion is of the form

$(a-b)(a^{n-1}{b}^0+a^{n-2}{b}^1+a^{n-3}{b}^2+.....+a^0{b}^{n-1})$