Let $a$ ne a nonzero integer, p a prime, n a positive integer, and p does not divide n. Prove that $p | \Phi_{n}(a)$ if and only if a has period n in $(\mathbb{Z}/p\mathbb{Z})^{*}$
Actually I have almost done, I know how to prove from the right to the left, but when I do from the left to the right, I can not show that for any $d < n$ is a factor of $n$, $a^{d}$ is not $1$ in $(\mathbb{Z}/p\mathbb{Z})^{*}$. To prove this, I think we must use the condition p does not divide n, but don't knwo how.
Any hint is appreciated, thank you.