I have to prove that for any natural number $n$ there exists $i>0$ such that $n\mid F_i$, where $F_i$ is the $i$-th Fibonacci number.
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3That follows from $F_0=0$ and the fact that the Fibonacci sequence $\pmod{n}$ is periodic. – Jack D'Aurizio Apr 08 '15 at 18:18
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Deeply related: http://math.stackexchange.com/questions/872071/fibonacci-number-that-ends-with-2014-zeros/872077#872077 – Jack D'Aurizio Apr 08 '15 at 18:20
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As to how to prove periodicity, consider that you can run the sequence 'backwards' - given $F_{i+1}$ and $F_{i+2}$ you can find $F_i$, so if you can show that the sequence ${F_k}$ repeats some of its values then it must repeat all of its values. – Steven Stadnicki Apr 08 '15 at 18:20