SILK-ROAD 2005 Problem 1

Let $n \geq 2$ be natural number. Prove, that $(1^{n-1}+2^{n-1}+....+(n-1)^{n-1})+1$ divided by $n$ iff for any prime divisor $p$ of $n$ $p| \frac{n}{p}-1$ and $(p-1)| \frac{n}{p}-1$ .