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  "body": "\n## Definition\n\nLet $n$ be a positive integer and let $a$ be an integer.\nThe residue class of $a$ modulo $n$ is the set $\\overline{a} = \\{ b \\in \\mathbb{Z} : b \\equiv a \\pmod n \\}$ of the integers [[congruent|0TPW25CBN6]] to $a$ modulo $n$.\nThus $\\overline{a} = \\{ a + k n : k \\in \\mathbb{Z} \\} = a + n \\mathbb{Z}$.\n\n## Remarks\n\nTwo residue classes $\\overline{a}$ and $\\overline{b}$ are equal if $a \\equiv b \\pmod n$, and they have no common element otherwise.\nThere are exactly $n$ residue classes modulo $n$: $\\overline{0}, \\overline{1}, \\dots, \\overline{n - 1}$.\nOther notations are $[a]$, $[a]_n$ and $a \\bmod n$.\nThe residue classes are the elements of the ring of [[integers modulo n|01N3AYAA6N]].\n\n## Explanations\n\n### L1 human\n\nModulo $3$, each integer falls in one of three boxes: the multiples of $3$, the numbers $1$ more than a multiple of $3$, and the numbers $2$ more.\nEach box is a residue class. For example, $\\overline{1} = \\{ \\dots, -5, -2, 1, 4, 7, \\dots \\}$.\n",
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