Fibonacci gcd reduction ($\gcd (F_m,F_n)=\gcd (F_{m-n},F_n),m\ge n$)

Consecutive Fibonacci numbers are coprime by the Euclidean algorithm. The Fibonacci addition formula writes $F_m=F_{m-n}{F}_{n+1}+F_{m-n-1}{F}_n$ for $m>n$. Taking the greatest common divisor with $F_n$ removes the last summand, and multiplication by the coprime factor $F_{n+1}$ leaves that gcd unchanged. The case $m=n$ uses $F_0=0$. Applying the Euclidean algorithm to the indices yields the strong-divisibility identity $\gcd (F_m,F_n)=F_{\gcd (m,n)}$ for nonnegative indices, with the usual zero conventions.