2019 AMC 12A 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 \qquad\textbf{(B) } 14 \qquad\textbf{(C) } 15 \qquad\textbf{(D) } 16 \qquad\textbf{(E) } 17

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.