TITLE: Show that a prime p divides $a^p - a$.
QUESTION [0 upvotes]: Let $p$ be a prime. I want to show that $p | a^p - a$. I want to use induction. 
I showed that this is true for the case $a = 1$. I'm having trouble with the bridge.

REPLY [2 votes]: Hint: We assumed $a^p \equiv a \pmod{p}$
Consider the Binomial expansion of: $(a+1)^p$ : 
$(a+1)^p = a^p + {p \choose 1} a^{p-1} + {p \choose 2} a^{p-2} + \cdots + {p \choose p-1} a + 1.$
Now see how you can reduce this expression into something much simpler, where you can apply the assumption. You may not use the assumption until the end. 
(comment if you would like further help!)
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