Fibonacci numbers. how to prove?

Question

We are give Fibonacci numbers.{fi | i ∈ N}, where f0 = 0, f1 = 1, fn+2 = fn +fn+1, n∈ N. How to proof with mathematical induction that if n divides by m, then fn divides by fm? I am having trouble with thinking, what should be the transition.

Answer 1

First prove by induction on $m$ that

$F_{m+n}=F_{m+1}F_n+F_mF_{n+1}-F_m F_n$

Then put $m=kn$ and find that if $F_n|F_{kn}$ then $F_n|F_{(k+1)n}$.