powers.......

What is the remainder when $13^{51}$ is divided by 5?

13^51 mod 5 =2 Remainder.

What is the remainder when $13^{51}$ is divided by 5?

$\begin{array}{|rcll|} \hline && 13^{51} \pmod {5} \qquad &| \qquad 13 \pmod 5 \equiv 3 \\ &\equiv& 3^{51} \pmod {5} \qquad &| \qquad 3^4 \equiv 1 \pmod 5 \\ &\equiv & 3^{4\cdot 12+3} \pmod {5} \\ &\equiv & 3^{4\cdot 12} \cdot 3^3 \pmod {5} \\ &\equiv & (3^{4})^{12} \cdot 3^3 \pmod {5} \qquad &| \qquad 3^4 \equiv 1 \pmod 5 \\ &\equiv & (1)^{12} \cdot 3^3 \pmod {5} \qquad &| \qquad (1)^{12}=1\\ &\equiv & 1 \cdot 3^3 \pmod {5} \\ &\equiv & 27 \pmod {5}\\ &\equiv & 2 \pmod {5}\\ \hline \end{array}$