Proof Writing & Problem Solving · Superchallenge problem

With $F_0 = 0$, $F_1 = 1$ and $F_{n+1} = F_n + F_{n-1}$, induction on $n$ proves $F_{m+n} = F_m F_{n+1} + F_{m-1} F_n$, and also that consecutive Fibonacci numbers are coprime. Use these to find $\gcd(F_{45}, F_{60})$.