How to determine congruence manually How is it possible to determine if the the following congruence is true manually?
$$
2015^{53} \equiv 8 \pmod{11}
$$

Since $2015 = 11 \cdot 183 + 2$, then $2015 \equiv 2 \pmod{11}$.
Our problem has now been reduced to showing that $2^{53} \equiv 8 \pmod{11}$.
Since $2^{53} = \left(2^5\right)^{10} \cdot 2^3 \pmod{11}$ and $2^5 = 32\equiv -1 \pmod{11}$ then $$2^{53} \equiv (-1)^{10} \cdot 2^3 \pmod{11} = 8 \pmod{11}$$
