Problem

Source: Chinese TST

Tags: modular arithmetic, number theory, relatively prime, number theory proposed



Let $ n > 1$ be an integer, and $ n$ can divide $ 2^{\phi(n)} + 3^{\phi(n)} + \cdots + n^{\phi(n)},$ let $ p_{1},p_{2},\cdots,p_{k}$ be all distinct prime divisors of $ n$. Show that $ \frac {1}{p_{1}} + \frac {1}{p_{2}} + \cdots + \frac {1}{p_{k}} + \frac {1}{p_{1}p_{2}\cdots p_{k}}$ is an integer. ( where $ \phi(n)$ is defined as the number of positive integers $ \leq n$ that are relatively prime to $ n$.)