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  "body": "\n## Definition\n\nLet $n$ be a positive integer and let $a$ be an integer.\nA modular multiplicative inverse of $a$ modulo $n$ is an integer $x$ such that $a x \\equiv 1 \\pmod n$.\n\n## Remarks\n\nAn inverse of $a$ modulo $n$ exists if and only if $\\gcd(a, n) = 1$. See [[existence of inverses modulo n|0M67FWZWPQ]].\nIf $x$ is an inverse of $a$, then the inverses of $a$ are exactly the integers $y \\equiv x \\pmod n$. So the inverse is unique as a residue class, and we write it as $a^{-1} \\bmod n$.\nIn the ring of [[integers modulo n|01N3AYAA6N]], the class $\\overline{x}$ is the inverse of $\\overline{a}$.\nFor example, $3^{-1} \\equiv 5 \\pmod 7$, because $3 \\cdot 5 = 15 \\equiv 1 \\pmod 7$.\n\n## Explanations\n\n### L1 human\n\nModulo $n$, the inverse of $a$ plays the part of the fraction $1/a$.\nTo solve $3 x \\equiv 4 \\pmod 7$, multiply both sides by $5$, the inverse of $3$. You get $x \\equiv 20 \\equiv 6 \\pmod 7$.\n",
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