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Introduction

This is a beautiful theorem which states that the difference between a positive integer $a$ and a prime power of $a$, $a^p-a$, is divisible by $p$. It’s history can be found in Wikipedia: Fermat posited the theorem (and as usual, did not give the proof) while Euler first published a proof using mathematical induction. Below, we will state the theorem and provide a simple-to-understand proof using only modular arithmetic, followed by another simple proof using mathematical induction.