Number theory basic

Let $S$ be the given sum. We need to find $S \bmod 10$ .

$\begin{aligned} S & \equiv \sum_{k=1}^{2020} k^k \text{ (mod 10)} \\ & \equiv \sum_{k=1}^{2020} \left(10^{k-1}+k\right)^{10^{k-1}+k} \text{ (mod 10)} \\ & \equiv 202 \sum_{k=1}^{10} k^k \text{ (mod 10)} \\ & \equiv 2 \left(1^1+2^2+3^3+4^4+5^5+6^6+7^7+8^8+9^9+10^{10}\right) \text{ (mod 10)} \\ & \equiv 2 \left(1+4+7+6+5+6+3+6+9+0\right) \text{ (mod 10)} \\ & \equiv 2(7) \equiv \boxed 4 \text{ (mod 10)} \end{aligned}$