Let F0 = 0, F1 = 1, F2 = 1, . . ., F99 be the first 100 Fibonacci numbers (recall that Fn = Fn−1 + Fn−2 for n ≥ 2).

Question

Let F0 = 0, F1 = 1, F2 = 1, . . ., F99 be the first 100 Fibonacci numbers (recall that Fn = Fn−1 + Fn−2 for n ≥ 2). how many of them are divisible by 3

Answer 1

$$\begin{align*} F_0 &\equiv 0 \mod{3} \\ F_1 &\equiv 1 \mod{3} \\ F_2 &\equiv 1 \mod{3} \\ F_3 &\equiv 2 \mod{3} \\ F_4 &\equiv 0 \mod{3} \\ F_5 &\equiv 2 \mod{3} \\ F_6 &\equiv 2 \mod{3} \\ F_7 &\equiv 1 \mod{3} \\ F_8 &\equiv 0 \mod{3} \text{ Pattern repeats here}\\ F_9 &\equiv 1 \mod{3} \\ F_{10} &\equiv 1 \mod{3} \\ F_{11} &\equiv 2 \mod{3} \\ F_{12} &\equiv 0 \mod{3} \\ \end{align*}$$