2020 AMC 10A Problem 21

Let $X=2^{17}$. Then $$\begin{gathered} \dfrac{2^{289}+1}{2^{17}+1} = \dfrac{X^{17}+1}{X+1} \\ = X^{16}-X^{15}+X^{14} \\ {}-\cdots-X+1 \end{gathered}$$. Pair consecutive terms: $X^{16}-X^{15}$, $X^{14}-X^{13}$, $\ldots$, $X^2-X$, and then the final $+1$. Each pair is $2^{17m}(2^{17}-1)$, contributing $17$ ones in binary. There are $8$ such pairs plus the final $1$, so $k=8\cdot17+1=137$. Thus, C is the correct answer.