Is Fibonacci Sequence Unlucky?

First terms of Fibonacci sequence are: $1, 1, 2, 3, 5, 8, 13$ . So there is at least one. Next we take Fibonacci sequence modulo $13$ . If there are consequtive terms of the form $a, 0$ (and there are $8, 13$ ) then next terms are: $a, 0, a, a, 2a, 3a, 5a, 8a, 13a$ but $13a = 0$ modulo $13$ and since $8a, 0$ is of the form $b, 0$ $(b$ here is just some value e.g. $8a$ ) we obtain by induction that there are infinitely many terms divisible by $13$ .