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?

Another way is by induction:
$$
2^1+1 = 3 = 3 \cdot 1
$$
Then, if $2^k+1 = 3j, j \in \mathbb{N}$, then
\begin{align}
2^{k+2}+1 & = 4\cdot2^k+1 \\
          & = 4(2^k+1)-3 \\
          & = 4(3j)-3 \qquad \leftarrow \text{uses induction hypothesis} \\
          & = 3(4j-1)
\end{align}
