Proof Writing & Problem Solving · Superchallenge problem

A draft uses the lemma "if $n^2 \equiv 1 \pmod m$, then $n \equiv 1$ or $n \equiv -1 \pmod m$." It is true for $m = 5$ but false for $m = 8$, since $3^2 = 9 \equiv 1 \pmod 8$. For how many integers $m$ with $2 \le m \le 50$ is the lemma true for every integer $n$?