Calculate GCD(Fib(531185354674),Fib(613570636967)) 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.

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 $F_n$.

