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title: 1. Property
tags: [maths, S1, fiche]

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###### tags: `fiche` `S1` `maths`
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  <span>Version 1.0.0</span>
  <span>2020/12/04</span>
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	<h1 style="font-size:3.5em;line-height: 2.5;">S1 Maths - Fermat's Little Theorem : Cours</h1>
	<div>
          Adam Alani <a href="mailto:adam.alani@epita.fr">&lt;adam.alani@epita.fr&gt;</a> <br>
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# 1. Property


Let $p$ be a prime number $\in \mathbb Z$

We have: $a^p \equiv a [p]$

## 1.1 Corollary:

Let $p$ be a prime number, $a \in Z$ such that $gcd(p , a) = 1$

Example

$8^{30} \equiv 1[31]$

$8^{31-1} \equiv 8^{30}$       and 31 is a prime number then $gcd(31, 8) = 1$

Hence:

$8^{31-1} \equiv 1[31]$

$8^{31} \equiv 8[31]$

## 1.2. Demo

$p$ a prime number, $a \in z$ and $gcd ( a , p) = 1$

$$
\begin{align}
&\iff a^p \equiv a[p] \\
&\iff p | a^p - a \\
&\iff p | a(a^{p-1} - 1) \text { and } gcd(a , p) = 1 \text{ ( gauss) } \\
&\iff p | a^{p-1} \equiv 0[0] \\
&\iff a^{p-1} \equiv 1[p] \\
\end{align}
$$