Show that if n divides a single Fibonacci number, then it will divide infinitely many Fibonacci numbers.
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.
Found the alternative form: $\sum _{k=1}^\infty ((1-\frac{1}{k})^{k})^k$. Tried various criteria, no luck so far.