3. What is the largest integer $k$ such that $k+1$ divides
$$
k^{2020}+2 k^{2019}+3 k^{2018}+\cdots+2020 k+2021 ?
$$
Answer: 1010
Solution: The remainder when the polynomial $x^{2020}+2 x^{2019}+3 x^{2018}+\cdots+2020 x+2021$ is divided by $x+1$ is
$$
(-1)^{2020}+2(-1)^{2019}+3(-1)^{2018}+\cdots+2020(-1)+2021=1010(-1)+2021=1011 .
$$

Therefore, $k+1$ divides $k^{2020}+2 k^{2019}+3 k^{2018}+\cdots+2020 k+2021$ precisely when $k+1$ divides 1011 . The largest $k$ for which this is true is 1010 .