number theory

What is the remainder when $2013^{13}$ is divided by $13$?

$\begin{array}{|rcll|} \hline && \mathbf{2013^{13} \pmod{13}} \quad &| \quad 2013 \equiv 11 \pmod{13} \\ &\equiv&11^{13}\pmod{13} \quad &| \quad 11^4 \equiv 3 \pmod{13} \\ &\equiv&\left( 11\right)^{4*3+1} \pmod{13} \\ &\equiv& \left( 11^4 \right)^3*11 \pmod{13} \\ &\equiv& \left( 3 \right)^3*11 \pmod{13} \\ &\equiv& 27*11 \pmod{13}\quad &| \quad 27 \equiv 1 \pmod{13} \\ &\equiv& 1*11 \pmod{13} \\ &\equiv& \mathbf{ 11 \pmod{13} } \\ \hline \end{array}$

The reminder is 11