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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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: