I am looking for a concise proof that the length of the smallest period of the sequence $$f[n] = a n \pmod N $$ is $N$ if $(a,N) = 1$. From the Pigeonhole Principle, it is not hard to show that $f[n]$ is periodic with $N$, but how do I know that it is not periodic with a number $M < N$. Is a proof by contradiction the way to go?
I found a lemma in the proof of Fermat's Theorem here, that takes me most of the way by arguing that of $1a, 2a, 3a, \ldots, (N-1)a$ are not congruent modulo $N$ if $(a,N)=1$. This question seems so trivial and is used in discussions of the periodicity of sequences so I thought there might be several ways to show it to be true.