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  "body": "\n## Statement\n\nLet $m$ and $n$ be positive integers, and let $a$ and $b$ be integers. Let $g = \\gcd(m, n)$ and $M = \\operatorname{lcm}(m, n)$.\nThe system $x \\equiv a \\pmod m$, $x \\equiv b \\pmod n$ has an integer solution if and only if $a \\equiv b \\pmod g$.\nIn this case, two solutions are congruent modulo $M$.\n\n## Remarks\n\nWhen $g = 1$, this is the [[Chinese remainder theorem|0FNN9B4W64]] for two moduli.\nThe condition is necessary: a solution $x$ gives $a \\equiv x \\equiv b \\pmod g$, since $g$ divides $m$ and $n$.\nFor $k$ moduli $n_1, \\dots, n_k$, the system $x \\equiv a_i \\pmod{n_i}$ has a solution if and only if $\\gcd(n_i, n_j) \\mid a_i - a_j$ for all $i \\neq j$.\nFor example, $x \\equiv 1 \\pmod 4$ and $x \\equiv 3 \\pmod 6$ has the solutions $x \\equiv 9 \\pmod{12}$. But $x \\equiv 1 \\pmod 4$ and $x \\equiv 2 \\pmod 6$ has no solution.\n\n## Explanations\n\n### L1 human\n\nWhen the moduli share a factor $g$, the two conditions both say something about $x$ modulo $g$.\nThey must agree there. If they agree, there is a solution, and it is unique modulo the lcm.\n",
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