AMC 12 2019a Problem 11

For some positive integer $k$, the repeating base-$k$ representation of the (base-ten) fraction $\frac{7}{51}$ is $0.\overline{23}_k = 0.232323..._k$. What is $k$?

$\textbf{(A) } 13 \textbf{(B) } 14 \textbf{(C) } 15 \textbf{(D) } 16 \textbf{(E) } 17$