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Harvard-MIT Mathematics Tournament

Algebra Difficulty 5.0 AIME Prove it United States

Problem:

Define the Fibonacci numbers by $F_{0}=0$ , $F_{1}=1$ , $F_{n}=F_{n-1}+F_{n-2}$ for $n \geq 2$ . For how many $n$ , $0 \leq n \leq 100$ , is $F_{n}$ a multiple of $13$ ?

Solution

Solution:

The sequence of remainders modulo $13$ begins $0, 1, 1, 2, 3, 5, 8, 0$ , and then we have $F_{n+7} \equiv 8 F_{n}$ modulo $13$ by a straightforward induction. In particular, $F_{n}$ is a multiple of $13$ if and only if $7 \mid n$ , so there are $15$ such $n$ .

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