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  "body": "\n## Statement\n\nLet $n \\geq 1$ and $a$ be integers.\nThe residue class $[a]$ is a unit of the ring $\\mathbb{Z}/n\\mathbb{Z}$ if and only if $\\gcd(a, n) = 1$.\n\n## Remarks\n\nIf $\\gcd(a, n) = 1$, then Bézout's lemma gives integers $x$ and $y$ with $a x + n y = 1$. So $[a][x] = [1]$.\nIf $[a][x] = [1]$, then $a x - 1$ is a multiple of $n$, so each common divisor of $a$ and $n$ divides $1$.\nThe condition does not depend on the representative, because $a \\equiv b \\pmod{n}$ implies $\\gcd(a, n) = \\gcd(b, n)$.\nFor $n = 12$, the units are $[1]$, $[5]$, $[7]$ and $[11]$.\n\n## Explanations\n\n### L1 human\n\nA residue has a multiplicative inverse modulo $n$ exactly when it has no common factor with $n$.\nBézout's lemma gives the inverse.\nSo the [[group of units of Z/nZ|0F8X8ZRQ53]] is the set of residues coprime to $n$.\n",
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