Divisibility and the Fibonacci sequence While studying the Fibonacci sequence I encountered this problem in the handout, and I can not understand how to do it.

Show that if the Fibonacci sequence has a term divisible by a natural number $m$, then there are infinitely many such terms.


The duplication identity for the Fibonacci numbers states:
$$
    F_{2m} = F_m \left( F_m + 2 F_{m-1} \right)
$$
Hence, for some $k \in \mathbb{N}$, if $F_m$ is divisible by $k$, every element of the sequence $F_{2m}, F_{4m}, \ldots, F_{2^r m}, \ldots$ is also  divisible by $k$.
