I am working on a review problem and can't figure out how to go about getting to an answer. We are told to let $F_n$ be the $nth$ Fibonacci number (defined as $F_1=F_2=1,F_{n+1}=F_n+F_{n-1}$). Show that, for any positive integer a, there is some $N>0$ such that $F_N $is divisible by a. Any help with this problem would be greatly appreciated. Thank you!
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2Consider the Fibonacci numbers modulo $a$. At any point if you know two successive numbers in the sequence you can work both forward and backward uniquely. And there are only $a^2$ possible such pairs, so sooner or later you will reach a pair that you have seen before. Because each pair has only one predecessor, this means that the initial pair $(1,1)$ is part of a cycle, and immediately before that comes the pair $(0,1)$, which corresponds to a number divisible by $a$. – hmakholm left over Monica May 14 '15 at 02:02
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@Henning Makholm What is the "initial pair" and why is it (1,1)? – dalastboss May 14 '15 at 02:05
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1@dalastboss: $(F_1,F_2)\equiv(1,1)\pmod a$. – hmakholm left over Monica May 14 '15 at 02:08
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Oh okay, gotcha, thanks – dalastboss May 14 '15 at 02:15
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Don't reinvent the wheel (cycle) - see here and see here for hints. – Bill Dubuque May 14 '15 at 02:32