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  "body": "\n## Statement\n\nLet $n$ be a positive integer, and let $a$, $b$ and $c$ be integers. Let $d = \\gcd(c, n)$.\nIf $c a \\equiv c b \\pmod n$, then $a \\equiv b \\pmod{n / d}$.\n\n## Remarks\n\nIn particular, if $c$ and $n$ are [[coprime|0GFKP8F8AG]], then $c a \\equiv c b \\pmod n$ implies $a \\equiv b \\pmod n$.\nThe modulus can become smaller. For example, $2 \\cdot 1 \\equiv 2 \\cdot 4 \\pmod 6$, but $1 \\not\\equiv 4 \\pmod 6$. Here $d = 2$, and indeed $1 \\equiv 4 \\pmod 3$.\nThe proof: $n \\mid c (a - b)$ gives $\\frac{n}{d} \\mid \\frac{c}{d} (a - b)$. Since $n / d$ and $c / d$ are coprime, the [[generalized Euclid's lemma|03BAREW1Z5]] gives $\\frac{n}{d} \\mid a - b$.\n\n## Explanations\n\n### L1 human\n\nYou can divide both sides of a congruence by a number that has no common factor with the modulus.\nIf the number shares a factor with the modulus, you must also divide the modulus by that common factor.\n",
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