$2^{2n+1} +1$ divisible by $3$ $2^{2n+1} +1$ divisible by $3$
Inducción over n
Let $n=1$, then: $$2^{2(1)+1}+1=2^{3}+1=9$$ 
It works!
Hip. $$2^{2n+1} +1=3k$$
Then we have to show: $$2^{2n+3} +1=3k$$
Any guess? 

HINT
$$2^{2n+3} +1=2^2 \cdot 2^{2n+1} +1\stackrel{Hyp.}=2^2(3k-1)+1$$
