$p|a^{p-1}-1$, can I prove that $p\nmid a$?

Question

Trying to understand the connections between concepts.

If p is prime and a is a positive integer, then if

$p|a^{p-1}-1$, can I prove that $p\nmid a$?

Answer 1

Suppose $p | a$. Then $p | a^{p-1}$ too. But then, $a^{p-1} - 1 \equiv (p-1) {\rm mod \ } p$, so $ p \nmid a^{p-1} - 1$.

Answer 2

If $p \mid a$ then $p \mid a^{p-1}$, so if $p \mid a^{p-1}-1$, then $p \mid -1$, a contradiction.