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  "body": "\n## Summary\n\nThe Euclidean algorithm computes the [[greatest common divisor|02WF878BH1]] of two natural numbers $a$ and $b$.\nIt uses two rules: $\\gcd(a, 0) = a$, and $\\gcd(a, b) = \\gcd(b, a \\bmod b)$ when $b \\neq 0$.\nIt applies the second rule until the second number is $0$. Then the first number is the result.\nThe algorithm stops, because the second number gets smaller at each step.\n\n## Remarks\n\nThe second rule is true because $a$ and $b$ have the same common divisors as $b$ and $a \\bmod b$.\nThe number of steps is at most a constant times the number of digits of the smaller input. Lamé's theorem gives this bound.\nThe algorithm also works in each Euclidean domain, for example in the ring of polynomials over a field.\n\n## Explanations\n\n### L1 human\n\nTo find $\\gcd(48, 18)$, divide again and again, and keep the divisor and the remainder.\n$48 = 2 \\cdot 18 + 12$, then $18 = 1 \\cdot 12 + 6$, then $12 = 2 \\cdot 6 + 0$.\nThe last nonzero remainder is $6$, so $\\gcd(48, 18) = 6$.\n",
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