{
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  "body": "\n## Definition\n\nLet $a$ and $n$ be integers with $n \\neq 0$.\nThe integer $a \\bmod n$ is the remainder $r$ of the [[division of $a$ by $n$|0CHQKZFK5S]].\nThus $a \\bmod n$ is the unique integer $r$ such that $0 \\le r \u003c |n|$ and $a - r$ is a multiple of $n$.\n\n## Remarks\n\nFor example, $17 \\bmod 5 = 2$ and $(-17) \\bmod 5 = 3$.\nMany programming languages use a different convention. In them, the remainder has the sign of $a$, so the result for $-17$ and $5$ is $-2$.\nThe Lean library Mathlib sets $a \\bmod 0 = a$.\n\n## Explanations\n\n### L1 human\n\nThe value $a \\bmod n$ is what is left after you take away from $a$ as many copies of $n$ as you can.\nA clock uses this operation: $15$ hours after $0$ o'clock, a $12$-hour clock shows $15 \\bmod 12 = 3$.\n",
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