Ok to mod exponents when solving modular expressions with integer exponents? [duplicate]
Since $12\equiv -1 \bmod 13$, for any odd exponent $12^{2n+1} \equiv -1 \bmod 13$ and for any even exponent $12^{2n} \equiv 1 \bmod 13$. You just have to see that the exponent in your expression must be even and you are done.