Fibonacci Day: Divide in all the Primes. Each and every one

I don't know if you noticed, but today is a Fibonacci day. It is April 21, and 21 is the eighth Fibonacci number.  Let's keep on the topic of prime numbers, and talk about a really cool correlation between the two sequences.

Take a prime number, like 11. Is eleven a factor of a Fibonacci number? Well, yes, it is. It divides the number 55.

What about 23? It divides 46368, the twenty-fourth Fibonacci number.

Take 101, a big prime. It actually divides the 100th Fibonacci number, which is 354224848179261915075.

In fact, every single prime number divides a Fibonacci number. Unfortunately, I do not know the proof for this, but if you are familiar with it, please comment below and tell us. This statement can actually be made cooler, believe it or not.

Every prime number (that we will call P) ending in 1 or 9 is a factor of the Fibonacci number in position P-1. Furthermore, any prime ending in 3 or 7 is a factor of the Fibonacci number in position P+1.  I thought that even though I don't know the proof, it is pretty cool that this correlation exists nonetheless.