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  "body": "\n## Statement\n\nLet $p$ be a [[prime number|0SN0MAEKYX]] and let $k$ be a positive integer.\nThen $\\varphi(p^k) = p^{k-1} (p - 1) = p^k - p^{k-1}$, where $\\varphi$ is [[Euler's totient function|0VV04M41AN]].\n\n## Remarks\n\nThe proof: an integer is coprime to $p^k$ if and only if $p$ does not divide it. Among $1, \\dots, p^k$, the multiples of $p$ are $p, 2p, \\dots, p^{k-1} \\cdot p$, so there are $p^{k-1}$ of them.\nFor example, $\\varphi(8) = 4$ and $\\varphi(27) = 18$.\nWith the [[multiplicativity of the totient function|0FSJA9M43W]], this formula gives $\\varphi(n)$ from the prime factorization of $n$.\n\n## Explanations\n\n### L1 human\n\nAmong the numbers $1$ to $p^k$, exactly one in $p$ is a multiple of $p$.\nRemove these numbers. The fraction $1 - 1/p$ of the numbers stays, so $\\varphi(p^k) = p^k (1 - 1/p)$.\n",
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