AHSME 1961 Problem 27

Given two equiangular polygons $P_1$ and $P_2$ with different numbers of sides; each angle of $P_1$ is $x$ degrees and each angle of $P_2$ is $kx$ degrees, where $k$ is an integer greater than $1$. The number of possibilities for the pair $(x, k)$ is: $\textbf{(A)}\ \infty \textbf{(B)}\ \text{finite, but greater than 2} \textbf{(C)}\ 2 \textbf{(D)}\ 1 \textbf{(E)}\ 0$