Selection tests for the Gulf Mathematical Olympiad 2013

Let $m$ be a positive integer and consider the infinite set of pairs $(F_k,F_{k+1})$, for $k\in \mathbb{N}$. By the pigeonhole principle, there exists a pair $(a,b)$ of integers $0\le a,b\le m-1$ and an infinite sequence of integers $0<k_1<k_2<\cdots$ such that $$(F_{k_i},F_{k_i+1})\equiv (a,b)\mathrm{mod}m,\text{ for all }i\ge 1.$$ Therefore $$(F_{k_i-1},F_{k_i})=(F_{k_i+1}-F_{k_i},F_{k_i})\equiv (b-a,a)\mathrm{mod}m,\text{ for all }i\ge 1.$$ We keep descending in this way until we get $$(F_{k_i-k_1+2},F_{k_i-k_1+3})\equiv (F_2,F_3)\equiv (1,2)\mathrm{mod}m,\text{ for all }i\ge 1.$$ Let $n_i=k_i-k_1+2$, for all $i\ge 1$. Clearly, the infinite sequence $2=n_1<n_2<\cdots$ is increasing and we have $$\begin{array}{c}{F}_{n_i}+2\equiv F_2+2\equiv 3\mathrm{mod}m,F_{n_i+1}+1\equiv F_3+1\equiv 3\mathrm{mod}m\\ \text{ and }{F}_{n_i+2}\equiv F_{n_i+1}+F_{n_i}\equiv F_3+F_2\equiv 3\mathrm{mod}m\end{array}$$