1322.11 – Giant Powers of 3

There is a pattern to the remainders when powers of 3 are divided by 13:

$$\begin{array}{cc}\textbf{Power } & \textbf{ Remainder} \\3^1 = 3 & 3 \\3^2 = 9 & 9 \\3^3 = 27 & 1 \\3^4 = 81 & 3 \\3^5 = 243 & 9 \\3^6 = 729 & 1\end{array}$$

And so it goes. We observe that the remainder is $1$ when the exponent is a multiple of $3$, $3$ when the exponent is $1$ more than a multiple of $3$, and $9$ when the exponent is $2$ more than a multiple of $3$. Now, $2020 = 3 \times 673 + 1$, i.e., $2020$ is $1$ more than a multiple of $3$. So $3^{2020}$ goes with $3^1$ and $3^4$. The remainder is $3$.