Let $gcd(m,n)=d$. Prove that for the Fibonacci numbers sequence, $gcd(f_m, f_n)=f_d$
This is in Chapter 7 of Introductory Combinatorics by Brualdi. Here is the answer provided at the end of the book, but I am not sure how to proceed:
Suppose $m=qn+r$, we know that $f_m=f_{qn-1}f_r+f_{qn}f_{r+1}$. Therefore, $gcd(f_m,f_n)=gcd(f_{qn-1}f_r, f_n)$. Then we can proceed by applying the standard algorithm for calculating GCDs.
How should we proceed specifically? Should we try to write $r$ as $an+b$, just like we did for $m$?
