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'Fibonacci Factors' printed from https://nrich.maths.org/
Make a list of Fibonnaci numbers and mark the even ones. Now $f_0$
is even and $f_1$ is odd so the sequence starts even, odd, odd,
even, ... Look for a pattern in the occurrence of even Fibonnaci
numbers in the sequence, then prove that your pattern must continue
indefinitely in the sequence.
Again look for a pattern in the occurrences of multiples of 3 in
the Fibonnaci sequence. To prove the pattern always applies use the
Fibonnaci difference relation $f_{n+2}=f_{n+1}+f_n$ repeatedly to
show that if a certain term is divisible by 3 then other terms
further along the sequence will also be divisible by 3.