How to find an odd prime factor of $F(2013)$?

Question

Here,

$F(0) = 0, F(1) = 1,$ and $F(n) = F(n − 1) + F(n − 2)$ for $n ≥ 2.$

Please give hints, not full answers.

Thanks!

Answer 1

Hints:

1. The $(kn)^{\text{th}}$ Fibonacci number is a multiple of $F(n)$, where $k, n \in \Bbb{N}$.

2. $2013 = 3 \cdot 11 \cdot 61$