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# Fermat's little theorem

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 Edit Definition/Summary Fermat's little theorem states that if $p$ is a prime number, then for any integer $a$, $a^{p}-a$ will be divisible by

 Edit Equations $$a^{p-1}\equiv1\pmod p \quad (\text{for\ }a \not\equiv 0 \pmod p)$$ $$a^p\equiv a\pmod p$$

 Edit Scientists Pierre de Fermat (1601?-1665)

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 Edit Breakdown Mathematics > Number Theory >> Congruences

 Edit Extended explanation Fermat's Little Theorem If p is a prime number and a an integer, then $$a^p\equiv a\ (p)$$ In order to prove Fermat's Little theorem, we will start by proving a superficially slightly weaker result, which is also referred to as Fermat's Little Theorem, on occasion. The two results imply each other, however. Theorem Let a and p be coprime, then $$a^{p-1}-1 \equiv 0\ (p).$$ Proof Start by listing the first p-1 positive multiples of a: $$a, 2a, 3a, \ldots, (p -1)a$$ Suppose that $ra$ and $sa$ are the same modulo $p$, with [itex]0