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  "body": "\n## Definition\n\nLet $n$ be a positive integer and let $a$ be an integer [[coprime|0GFKP8F8AG]] to $n$.\nThe multiplicative order of $a$ modulo $n$ is the smallest positive integer $k$ such that $a^k \\equiv 1 \\pmod n$.\nThe notation for it is $\\operatorname{ord}_n(a)$.\n\n## Remarks\n\nThe order exists, because $a^{\\varphi(n)} \\equiv 1 \\pmod n$ by [[Euler's theorem|0VC4DVE7DQ]].\nFor each integer $m \\ge 0$, $a^m \\equiv 1 \\pmod n$ if and only if $\\operatorname{ord}_n(a) \\mid m$. In particular, $\\operatorname{ord}_n(a) \\mid \\varphi(n)$.\nThe order of $a$ modulo $n$ is the order of $\\overline{a}$ in the group $(\\mathbb{Z}/n\\mathbb{Z})^\\times$.\nFor example, $\\operatorname{ord}_7(2) = 3$, because $2^1 = 2$, $2^2 = 4$ and $2^3 = 8 \\equiv 1 \\pmod 7$.\n\n## Explanations\n\n### L1 human\n\nCompute the powers $a, a^2, a^3, \\dots$ modulo $n$, until you reach $1$. The number of steps is the order.\nModulo $10$, the powers of $3$ are $3, 9, 7, 1$, so the order of $3$ modulo $10$ is $4$.\n",
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