How does one prove that if $n \mid (a^{n}-b^{n}) \ \Longrightarrow$ $ \displaystyle n \mid \frac{a^{n}-b^{n}}{a-b}$ where $a,b, n \in \mathbb{N}$.
What i thought of is to consider $(a-b)^{n} \equiv a^{n} + (-1)^{n}b^{n} \ (\text{mod} \ n)$ and if we suppose that $n$ is odd then we have, $(a-b)^{n} \equiv a^{n} -b^{n} \ (\text{mod} \ n)$ and since $n \mid (a^{n} - b^{n})$ we have $(a-b)^{n} \equiv 0 \ (\text{mod} \ n) $
I think i am far away from the conclusion of the problem, but this is what i could work on regarding the problem.