One of the most interesting divisibility properties of the Fibonacci numbers is
that for each prime p, there is a Fibonacci number Fn such that p divides Fn (see,
e.g. [5]). More specifically, for p 6= 5, p divides either Fp−1 if p ≡ ±1(mod 5), or
Fp+1 if p ≡ ±2(mod 5). For p = 5, one has of course p = Fp.