{
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  "body": "\n## Definition\n\nLet $n$ be a positive integer.\nA reduced residue system modulo $n$ is a set $R$ of integers with these properties.\nEach element of $R$ is [[coprime|0GFKP8F8AG]] to $n$.\nNo two distinct elements of $R$ are congruent modulo $n$.\nEach integer coprime to $n$ is congruent modulo $n$ to some element of $R$.\n\n## Remarks\n\nThus $R$ contains exactly one element of each [[residue class|0QRV8FPK5M]] of integers coprime to $n$. It has $\\varphi(n)$ elements, where $\\varphi$ is [[Euler's totient function|0VV04M41AN]].\nFor example, $\\{1, 5, 7, 11\\}$ is a reduced residue system modulo $12$.\nIf $R$ is a reduced residue system modulo $n$ and $\\gcd(a, n) = 1$, then $\\{ a r : r \\in R \\}$ is also a reduced residue system modulo $n$. This fact gives a proof of [[Euler's theorem|0VC4DVE7DQ]].\n\n## Explanations\n\n### L1 human\n\nTake a [[complete residue system|0E22JKS4W4]] modulo $n$, and keep only the numbers with no common factor with $n$.\nFor $n = 10$, the numbers $0, 1, \\dots, 9$ give the reduced residue system $\\{1, 3, 7, 9\\}$.\n",
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