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  "body": "\n## Definition\n\nLet $n$ be a positive integer.\nThe integers modulo $n$ form the set $\\mathbb{Z}/n\\mathbb{Z} = \\{ \\overline{0}, \\overline{1}, \\dots, \\overline{n - 1} \\}$ of the [[residue classes|0QRV8FPK5M]] modulo $n$.\nAddition and multiplication on $\\mathbb{Z}/n\\mathbb{Z}$ are $\\overline{a} + \\overline{b} = \\overline{a + b}$ and $\\overline{a} \\cdot \\overline{b} = \\overline{a b}$.\n\n## Remarks\n\nThe operations are well defined: the result does not depend on the choice of $a$ and $b$ in their classes. This follows from [[compatibility with addition|00FWBN7F0Y]] and [[compatibility with multiplication|072D5G4SK6]].\nWith these operations, $\\mathbb{Z}/n\\mathbb{Z}$ is a commutative ring with $n$ elements, with zero $\\overline{0}$ and identity $\\overline{1}$.\nOther notations are $\\mathbb{Z}_n$ and $\\mathbb{Z}/(n)$. The Lean library Mathlib writes `ZMod n`, and it sets `ZMod 0` to be $\\mathbb{Z}$.\nThe ring is a field if and only if $n$ is prime. See [[integers modulo a prime form a field|0R8YXRC9AJ]].\n\n## Explanations\n\n### L1 human\n\nIn $\\mathbb{Z}/12\\mathbb{Z}$ you compute as on a clock: $\\overline{9} + \\overline{5} = \\overline{14} = \\overline{2}$.\nProducts work the same way: $\\overline{5} \\cdot \\overline{7} = \\overline{35} = \\overline{11}$.\n",
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