How to prove that all odd powers of two add one are multiples of three 
For example
  \begin{align} 
2^5 + 1 &= 33\\
2^{11} + 1 &= 2049\ \text{(dividing by $3$ gives $683$)}
\end{align}

I know that $2^{61}- 1$ is a prime number, but how do I prove that $2^{61}+1$ is a multiple of three?

$2^2=4\equiv1\pmod 3$, so $4^k\equiv1\pmod3$ for all integers $k$.  And so for any odd number $2k+1$, we get $2^{2k+1}+1 = 4^k\cdot 2+1\equiv 2+1\equiv0\pmod3$.
