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  "body": "\n## Statement\n\nLet $p$ be a prime number, and let $a$ and $b$ be nonzero integers.\nThen $\\nu_p(a b) = \\nu_p(a) + \\nu_p(b)$, where $\\nu_p$ is the [[p-adic valuation|04VCPS30ZE]].\n\n## Remarks\n\nWith the convention $\\nu_p(0) = \\infty$, the formula holds for all integers $a$ and $b$.\nThe proof uses [[Euclid's lemma|051VYKA1EX]]. Write $a = p^i a'$ and $b = p^j b'$ with $p \\nmid a'$ and $p \\nmid b'$. Then $p \\nmid a' b'$, so $\\nu_p(a b) = i + j$.\nFor example, $\\nu_2(12 \\cdot 10) = \\nu_2(120) = 3 = 2 + 1$.\n\n## Explanations\n\n### L1 human\n\nThe factors $p$ in $a b$ come from $a$ or from $b$, and no new factor $p$ appears.\nSo the number of factors $p$ in the product is the sum of the numbers of factors $p$ in $a$ and in $b$.\n",
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