6. In the Cartesian coordinate system, a point whose both coordinates are integers is called an integer point. Let $k$ be an integer. When the intersection point of the lines $y=x-3$ and $y=k x+k$ is an integer point, the number of possible values for $k$ is ( ).
(A) 2
(B) 4
(C) 6
(D) 8
6.C.

From $\left\{\begin{array}{l}y=x-3, \\ y=k x+k,\end{array}\right.$ we solve to get $x=\frac{k+3}{1-k}=-1+\frac{4}{1-k}$. Therefore, $k=-3,-1,0,2,3,5$.