+26387 What is the remainder when $2^{2005}$ is divided by 7?
\(\begin{array}{|rcll|} \hline && 2^{2005} \pmod {7} \qquad &| \qquad 2^6 \equiv 1 \pmod 7 \\ &\equiv & 2^{6\cdot 334+1} \pmod {7} \\ &\equiv & 2^{6\cdot 334} \cdot 2 \pmod {7} \\ &\equiv & (2^{6})^{334} \cdot 2 \pmod {7} \qquad &| \qquad 2^6 \equiv 1 \pmod 7 \\ &\equiv & (1)^{334} \cdot 2 \pmod {7} \qquad &| \qquad (1)^{334}=1\\ &\equiv & 1 \cdot 2 \pmod {7}\\ &\equiv & 2 \pmod {7}\\ \hline \end{array} \)
