As explained in Excursion 4.5, the Fibonacci numbers are defined by the rules: F(0) = 0, F(1) = 1, and for all n with n ≥ 2, F(n) = F(n-1) + F(n-2). Which of these claims about the Fibonacci numbers is false? Select one:
a. For every n with n ≥ 5, F(n) ≥ 2n/2.
b. For every natural n, F(n) is divisible by 3 if and only if n is divisible by 4.
c. For every n, F(n) ≤ 2n.
d. The sum for i from 0 through n of F(i) is F(n+2) - 1.
This was a question on a test that I got wrong. My answer was b. I am looking to understand how this is possible. My understanding is that F(n) is a fibonacci number. If so, not all numbers are divisible by 3 and are also divisible by 4.
The statement doesn't say that. The statement says that if $F(n)$ is divisible by $3$, then $n$ is divisible by $4$, and vice versa.
To see that this statement is true, I advise you to think about the fact that $$F(n+1)\equiv F(n) + F(n-1)\mod 3$$
and then look at the whole sequence modulo $3$. You might notice a pattern.