$31 \cdot 3$ If the positive integer $k$ makes the equation $k x-12=3 k$ have an integer solution for $x$, then the number of possible values for $k$ is
(A) 3.
(B) 4.
(C) 5.
(D) 6.
(E) 7.
(43rd American High School Mathematics Examination, 1992)
[Solution] From the given, we have $x=\frac{3k+12}{k}=3+\frac{12}{k}$,
and the positive divisors of 12 are $1, 2, 3, 4, 6, 12$, making a total of 6, which results in 6 values of $k$ that make the equation have integer solutions.
Therefore, the answer is $(D)$.