If $m$ and $n$ are positive integers, then $(F_m,F_n)=F_{(m,n)}$. Edit: The $F$'s are Fibonacci numbers.
I need an idea on how to show the following:

If $m$ and $n$ are positive integers, then $(F_m,F_n)=F_{(m,n)}$.

I believe that using the fact that $F_{m+n}=F_mF_{n+1}+F_nF_{m-1}$ could come in handy. Moreover, Euclid's algorithm may as well be needed. But I am not certain, as there may be better methods to achieve this.
Thanks in advance.

As noted in the comments by sdcvvc, this answer to an earlier question completely answers this question as well.
