Let $a\in \mathbb Z$ and $\mathit\Phi_m(x)$.
Now I know that if $p$ is an odd prime less than $m$ and $p$ divides $\mathit\Phi_m(a)$ then $p$ divides $m$.
As $\mathit\Phi_m(a)$ divides $x^m-1$ in $\mathbb Z [x]$ is a stronger version of the above statement true, that is
If $p$ is an odd prime less than $m$ and $p$ divides $a^m-1$ then $p$ divides $m$
I couldn't find a valid argument to prove this nor was I able to find a counterexample.