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  "body": "\n## Statement\n\nLet $n$ be a positive integer and let $a$ be an integer.\nThen $a$ has a [[modular multiplicative inverse|00GXTQMY8Z]] modulo $n$ if and only if $a$ and $n$ are [[coprime|0GFKP8F8AG]].\n\n## Remarks\n\nIf $\\gcd(a, n) = 1$, [[Bézout's identity|0DQVHMYAGA]] gives integers $x$ and $y$ with $a x + n y = 1$. Then $a x \\equiv 1 \\pmod n$.\nConversely, if $a x \\equiv 1 \\pmod n$, then $a x + n y = 1$ for some integer $y$, so each common divisor of $a$ and $n$ divides $1$.\nThe [[extended Euclidean algorithm|0964YQFSA8]] computes the inverse.\nFor example, $4$ has no inverse modulo $6$, because $\\gcd(4, 6) = 2$.\n\n## Explanations\n\n### L1 human\n\nYou can \"divide by $a$\" modulo $n$ exactly when $a$ and $n$ have no common factor.\nModulo $10$, the numbers $1, 3, 7, 9$ have inverses, since $1 \\cdot 1$, $3 \\cdot 7$ and $9 \\cdot 9$ all end with the digit $1$.\nThe numbers $2, 4, 5, 6, 8$ have no inverse modulo $10$.\n",
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