1. ## Fibonacci numbers.

The Fibonacci numbers are defined by $F_0=0$, $F_1=1$ and $F_n=F_{n-1}+F_{n-2}$
Prove if $2|F_n$, then $3|n$

This is one of those times where it's easy to see intuitively, it's hard to see formally.
Considering the Fibonacci sequence
$0,1,1,2,3,5,8,13,21,36...$

It falls into a pattern of
$E,O,O,E,O,O,E,O,O,E,...$

From, as the first term $F_0$ is even, it's clear to see that every third term beginning from $F_1$ is divisible by 3, as it is the sum of the 2 previous numbers which are odd.

But this just scaffolding, scrap work.
Any tips on how to codify this formally?

2. Given the recursive nature of Fibonacci numbers, a proof by induction seems to be the logical way to go. Have you considered that?