Determine which Fibonacci numbers are even (a) Determine which Fibonacci numbers are even. Use a form of mathematical induction to prove your conjecture.
(b) Determine which Fibonacci numbers are divisible by 3. Use a form of mathematical induction to prove your conjecture
I understand that for part a that all multiples of 3 of n are even. So F(0),F(3),F(6)... I just don't understand how to prove it.
For part B it is the same thing except multiples of 4
Please help, thank you!

Instead of trying to write proofs of three things --- that it's even when the index is a multiple of $3$, odd when it's $1$ more than that, and odd when it's two more than that --- write just one proof that covers all three.
Modulo $2$ you start with $1,1,0$.  You've added the two $1$s to get $0$.
Then add $1$ and $0$ to get $1$: ${}\qquad 1,1,0,1$.
Then add $0$ and $1$ to get $1$: ${}\qquad 1,1,0,1,1$,
Then add those last two $1$s to get $0$: ${}\qquad 1,1,0,1,1,0$.
So if a sequence of three consecutive terms is $1,1,0$, then the next sequence of three consecutive terms is $1,1,0$.
