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  "body": "\n## Definition\n\nLet $p$ be a [[prime number|0SN0MAEKYX]] and let $n$ be a nonzero integer.\nThe $p$-adic valuation $\\nu_p(n)$ is the largest natural number $k$ such that $p^k \\mid n$.\nBy convention, $\\nu_p(0) = \\infty$.\n\n## Remarks\n\nFor a positive integer $n$, $\\nu_p(n)$ is the exponent of $p$ in the [[prime factorization|0AEKDFCYFA]] of $n$. It is $0$ when $p \\nmid n$.\nFor example, $\\nu_2(40) = 3$ and $\\nu_5(40) = 1$, because $40 = 2^3 \\cdot 5$.\nFor a nonzero rational number $a / b$, one sets $\\nu_p(a / b) = \\nu_p(a) - \\nu_p(b)$.\nThe Lean library Mathlib sets $\\nu_p(0) = 0$ for its function on natural numbers.\n\n## Explanations\n\n### L1 human\n\nThe number $\\nu_p(n)$ counts how many times you can divide $n$ by $p$ and still get an integer.\nFor $n = 72$ and $p = 2$: $72, 36, 18, 9$, so you can divide three times, and $\\nu_2(72) = 3$.\n",
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