Let $a$ and $n$ denote positive integers such that $n|a^n-1$. Prove that the numbers $a+1,a^2+2, \cdots a^n+n$ all leave different remainders when divided by $n$.
Source: India Postals 2015 Set 2
Tags: number theory, modular arithmetic
Let $a$ and $n$ denote positive integers such that $n|a^n-1$. Prove that the numbers $a+1,a^2+2, \cdots a^n+n$ all leave different remainders when divided by $n$.