KöMaL 2008 A459

Denote by $F_n$ the $n$th Fibonacci number ($F_0$=0, $F_1$=1, $F_{k+1}$=$F_k$+$F_{k-1}$). Show that one can find a positive integer $n$, having at least 1000 distinct prime divisors, such that $n$ divides $F_n$.