1992 AHSME Problem 10

The number of positive integers k for which the equation kx-12=3k has an integer solution for x is

\text{(A) } 3\quad \text{(B) } 4\quad \text{(C) } 5\quad \text{(D) } 6\quad \text{(E) } 7

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