Fibonacci multiple-index divisibility

For positive integers $m,n$, the Fibonacci numbers satisfy $F_m\mid F_{mn}$. The Fibonacci addition formula gives $F_{mn+m}=F_{m-1}{F}_{mn}+F_m{F}_{mn+1}$, so mathematical induction on $n$ proves the assertion. This explains why every fifth Fibonacci number is divisible by $F_5=5$.