Negative powers in modular arithmetic

Question

Suppose we have set $Z = \{0, 1, \dots, N-1\}$ with arithmetic operations modulo $N$; $a > 0$ is an element in $Z$.

Is it possible that $a^{-1}$ does not exist but $a^{-n}$ exists for some $n$, $1 < n < N$?

If this is impossible how to prove it?

Answer 1

Let $a^nb=1$. Then $a(a^{n-1}b)=1$.