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  "body": "\n## Statement\n\nLet $p$ be a [[prime number|0SN0MAEKYX]] and let $a$ be an integer.\nThen $a^p \\equiv a \\pmod p$.\n\n## Remarks\n\nIf $p \\nmid a$, multiply [[Fermat's little theorem|0M25PRN3WQ]] $a^{p-1} \\equiv 1 \\pmod p$ by $a$. If $p \\mid a$, both sides are $0$ modulo $p$.\nConversely, this form implies the form $a^{p-1} \\equiv 1$: cancel $a$, which is allowed when $p \\nmid a$.\nIn the ring $\\mathbb{Z}/p\\mathbb{Z}$, the form says that $x^p = x$ for each element $x$.\n\n## Explanations\n\n### L1 human\n\nThe form $a^p \\equiv a$ needs no hypothesis on $a$.\nFor example, with $p = 3$: $0^3 = 0$, $1^3 = 1$, $2^3 = 8 \\equiv 2$, and $4^3 = 64 \\equiv 1 \\equiv 4 \\pmod 3$.\n",
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