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  "body": "\n## Statement\n\nLet $m$ and $n$ be natural numbers, and let $F_k$ be the [[Fibonacci numbers|0GA9K57A39]].\nThen\n$$\\gcd(F_m, F_n) = F_{\\gcd(m, n)}.$$\n\n## Remarks\n\nExample: $\\gcd(F_{12}, F_{18}) = \\gcd(144, 2584) = 8 = F_6$.\nIn particular, $F_m$ divides $F_n$ when $m$ divides $n$. Consecutive Fibonacci numbers are coprime, because $\\gcd(n, n+1) = 1$ and $F_1 = 1$.\nThe proof follows the Euclidean algorithm and uses the [[addition formula|00CN2F997Q]].\n\n## Explanations\n\n### L1 human\n\nThe greatest common divisor of two Fibonacci numbers is again a Fibonacci number.\nTo find which one, take the greatest common divisor of the two indices.\n",
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