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Find the remainder when $5^{67}$ is divided by $11$.

$\begin{array}{|rcll|} \hline 5^{67} \pmod{11} &\equiv& 5^{5*13+2}\pmod{11} \\ &\equiv& \left(5^5 \right)^{13}5^2\pmod{11} \quad | \quad 5^5 \equiv 1\pmod{11} \\ &\equiv& 1^{13}5^2\pmod{11} \\ &\equiv& 5^2\pmod{11} \\ &\equiv& 25\pmod{11} \\ \mathbf{5^{67} \pmod{11}} &\equiv& \mathbf{ {\color{red}3}\pmod{11}} \\ \hline \end{array}$

The remainder is 3.