I am having some trouble with this proof and I need some help heading down the right direction:
Suppose $n = p_1p_2 \cdots p_k$ where $p_i$ are distinct primes and that $p_i - 1 \mid n - 1$. Show that $n$ is a Carmichael number, that is, that $a^{n - 1} \equiv 1 \pmod n$ for all $a$ with $\gcd(a, n) = 1$.
Thank you for your help.