Make a list of Fibonnaci numbers and mark the even ones. Now <i>f</i><sub>0</sub> is even and <i>f</i><sub>1</sub> 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.
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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 <i>f</i><sub><i>n</i>+2</sub>=<i>f</i><sub><i>n</i>+1</sub>+<i>f</i><sub><i>n</i></sub> 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.