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I have managed to show that $(a + b)^p \equiv a^p + b^p \pmod p$, $a$ and $b$ being any integer and $p$ any prime.

How can I prove from this that $a^p \equiv a \pmod p$?

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    This is Fermat’s so-called little theorem; you’ll find several proofs [here](http://en.wikipedia.org/wiki/Proofs_of_Fermat%27s_little_theorem). The one using the binomial theorem is probably the one that you want: use induction, taking $b=1$.2012-05-27
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    I've edited the title of your post to match better your question. Recommendation form here: [How can I ask a good question?](http://meta.math.stackexchange.com/questions/588/how-can-i-ask-a-good-question) *Make your title as descriptive as possible.*2012-05-27
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    @Martin: The trouble with title edits such as this is that it can make people think (if they don't look too closely) that the OP knew the name of the theorem and was just lazy about googling for a proof, whereas if you don't know the name, it's pretty hard to google.2013-02-19
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    @TaraB You are of course right, but I don't think that is such a big issue.2013-02-19

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