Make a list of Fibonnaci numbers and mark the even ones. Now f₀ is even and f₁ 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.