Prove for any integers \(m,n\ge0\)
that \(F_{m+n} = F_{m-1}F_n +
F_mF_{n+1}\).
Corollary: if \(k\mid
n\), then \(F_k\mid F_n\). This
can be proven by induction if we write \(n=sk\) for a natural \(s\), then \[F_{k+(s-1)k} = F_{k-1}F_{(s-1)k} +
F_kF_{(s-1)k+1}.\]