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  "body": "\n## Statement\n\nLet $p$ be a [[prime number|0SN0MAEKYX]], and let $a$ and $b$ be integers.\nIf $p \\mid a b$, then $p \\mid a$ or $p \\mid b$.\n\n## Remarks\n\nThe hypothesis that $p$ is prime is necessary. For example, $6 \\mid 4 \\cdot 9$, but $6 \\nmid 4$ and $6 \\nmid 9$.\nA short proof uses [[Bézout's identity|0DQVHMYAGA]]. If $p \\nmid a$, then $\\gcd(p, a) = 1$, so $p x + a y = 1$ for some integers $x$ and $y$. Multiply by $b$ to see that $p \\mid b$.\nThe [[generalized Euclid's lemma|03BAREW1Z5]] replaces the prime $p$ by an integer that is coprime to $a$.\nThe lemma is the main step in the proof of the uniqueness part of the [[fundamental theorem of arithmetic|084HH1NDCB]].\n\n## History\n\nThe lemma is Proposition 30 of Book VII of the Elements of Euclid.\n\n## Explanations\n\n### L1 human\n\nA prime cannot be split into smaller factors, so it cannot share itself between $a$ and $b$.\nIf $7$ divides a product $a b$, then $7$ already divides $a$ or $b$.\nFor example, $7 \\mid 14 \\cdot 3$ and $7 \\mid 14$.\n",
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