$\begin{array}{l}
2 + {2^2} + ... + {2^{10}}\\
= \left( {2 + {2^2}} \right) + \left( {{2^3} + {2^4}} \right) + ... + \left( {{2^9} + {2^{10}}} \right)\\
= 2\left( {1 + 2} \right) + {2^3}\left( {1 + 2} \right) + ... + {2^9}\left( {1 + 2} \right)\\
= 2.3 + {2^3}.3 + ... + {2^9}.3\\
= 3.\left( {2 + {2^3} + ... + {2^9}} \right)\,\,\, \vdots \,\,\,3\\
\Rightarrow A\,\,\, \vdots \,\,\,3
\end{array}$