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I picked this problem from here http://mathalon.in/?page=show_problem.php&pid=106 . I don't know how to solve it accurately. The number seems too to be done by any computation method.

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2 Answers 2

up vote 3 down vote accepted

Try and compute $\gcd(F_i,F_j)$ for some small $i$ and $j$ and see if you can spot a pattern.

Using the following three facts you can prove by induction that the pattern holds in general:

  • $\gcd(F_n,F_{n-1}) = 1$.
  • $F_{m+n} = F_{m+1}F_n + F_mF_{n-1}$.
  • $m$ divides $n \implies F_m$ divides $F_n$.
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GCD(Fib(531185354674),Fib(613570636967)) = Fib(GCD(531185354674,613570636967))

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Yes, and $\text{GCD}(531185354674, 613570636967)=67$ and $\text{Fib}_{67}=44945570212853$ –  Listing Jan 24 '11 at 13:20

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