The Fibonacci recurrence (F1 = 1, F2 = 1 and Fn = Fn - 1 + Fn - 2) leads to an infinite sequence of numbers starting with

1, 1, 2, 3, 5,
8, 13, 21, 34, 55,
89, 144, 233, 377, 610, ...

Note that the 5th, 10th and 15th numbers are all divisible by 5.
Show that every fifth number in the sequence is divisible by 5.

(In reply to modular solution by Charlie)

The process must repeat in any modulus.  I investigated the way it repeats in different moduli.  The results led me to some interesting sequences in the OEIS, but I couldn't think of a good Perplexus problem arising from it.

Posted by Jer on 2013-10-22 13:54:30