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  "body": "\n## Statement\n\nLet $m$ and $n$ be positive integers with $m \u003e n$.\nDivide $m$ by $n$ with the [[division algorithm|0CHQKZFK5S]]. This gives the unique integers $q$ and $r$ with $m = qn + r$ and $0 \\le r \u003c n$.\n\nIf $r \u003e 0$, then $\\gcd(m, n) = \\gcd(n, r)$, where $\\gcd$ is the [[greatest common divisor|0C8WYW4KYQ]].\n\nIf $r = 0$, then $n$ [[divides|096PW7Q014]] $m$, and $\\gcd(m, n) = n$.\n\n## Remarks\n\nThe first case lets us replace the pair $(m, n)$ by the smaller pair $(n, r)$ without changing the gcd.\nThe second case ends the process.\n",
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