Fibonacci divisibilty properties $ F_n\mid F_{kn},\,$ $\, \gcd(F_n,F_m) = F_{\gcd(n,m)}$ Can any one give a generalization of the following properties in a single proof? I have checked the results, which I have given below by trial and error method. I am looking for a general proof, which will cover the all my results below:


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*Every third Fibonacci number is even.

*3 divides every 4th Fibonacci number.

*5 divides every 5th Fibonacci number.

*4 divides every 6th Fibonacci number.

*13 divides every 7th Fibonacci number.

*7 divides every 8th Fibonacci number.

*17 divides every 9th Fibonacci number.

*11 divides every 10th Fibonacci number.

*6, 9, 12 and 16 divides every 12th Fibonacci number.

*29 divides every 14th Fibonacci number.

*10 and 61 divides every 15th Fibonacci number.

*15 divides every 20th Fibonacci number.


I guess that the standard way to understand all these divisibility results in one single swoop is to observe that the Fibonacci sequence modulo any number N becomes periodic.
For instance, Fibonacci modulo 2 is 0, 1, 1, 0, 1, 1, 0, ...... proving the even-ness of $F_n$ for $n=0,3,6,9,...$.
Fibonacci modulo $3$ is 0, 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2, 0, ..... making obvious that $3$ divides $F_n$ for $n=0, 4, 8, 12, ...$ .
Try yourself the next ones!
NOTE: the same technique can be applied to any linear recursive sequence with constant coefficients.
