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  "body": "\n## Statement\n\nLet $n$, $a$ and $b$ be integers.\nIf $n \\mid a b$ and $n$ and $a$ are [[coprime|0GFKP8F8AG]], then $n \\mid b$.\n\n## Remarks\n\nWhen $n$ is a prime $p$ and $p \\nmid a$, then $p$ and $a$ are coprime. So this result implies [[Euclid's lemma|051VYKA1EX]].\nThe proof uses [[Bézout's identity|0DQVHMYAGA]]: from $n x + a y = 1$ we get $b = n b x + a b y$, and $n$ divides both terms.\nFor example, $5 \\mid 3 \\cdot 10$ and $\\gcd(5, 3) = 1$, so $5 \\mid 10$.\n\n## Explanations\n\n### L1 human\n\nIf $n$ divides $a b$ but has no common factor with $a$, then all the factors of $n$ must come from $b$.\nFor example, $4 \\mid 3 b$ forces $4 \\mid b$, because $3$ gives no factor $2$.\n",
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  "msc": [
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  "name": "Generalized Euclid's lemma",
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