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  "aliases": [
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  "body": "\n## Definition\n\nLet $n$ be a positive integer.\nEuler's totient function $\\varphi(n)$ is the number of integers $k$ with $1 \\le k \\le n$ such that $k$ and $n$ are [[coprime|0GFKP8F8AG]].\n\n## Remarks\n\nFor example, $\\varphi(1) = 1$, $\\varphi(9) = 6$ and $\\varphi(12) = 4$. The integers coprime to $12$ in $\\{1, \\dots, 12\\}$ are $1, 5, 7, 11$.\nThe number $\\varphi(n)$ is the number of elements of a [[reduced residue system|0J0QEN1DZR]] modulo $n$.\nIf $n = p_1^{e_1} \\cdots p_r^{e_r}$ with distinct primes $p_i$, then $\\varphi(n) = n \\prod_{i=1}^{r} \\left(1 - \\frac{1}{p_i}\\right)$.\nThe Lean library Mathlib sets $\\varphi(0) = 0$.\n\n## Explanations\n\n### L1 human\n\nCount the numbers from $1$ to $n$ that have no common factor with $n$.\nFor $n = 10$, these are $1, 3, 7, 9$, so $\\varphi(10) = 4$.\n",
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