6. In the coordinate plane, a point whose both coordinates are integers is called an integer point. Let $I$ denote the set of all lines in the coordinate plane, $M$ denote the set of lines passing through exactly one integer point, $N$ denote the set of lines not passing through any integer point, and $P$ denote the set of lines passing through infinitely many integer points. Then the number of correct statements among (1) $M \cup N \cup P=I$, (2) $M \neq \varnothing$, (3) $N \neq \varnothing$, (4) $P \neq \varnothing$ is ( ).
A. 1
B. 2
C. 3
D. 4
6. D.

The line $y=\sqrt{2} x$ passes through exactly one integer point $(0,0)$, the line $y=\sqrt{2}$ does not pass through any integer points, and the line $y=x$ passes through infinitely many integer points. If the line $a x+b y+c=0$ passes through two integer points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$, then it passes through infinitely many integer points $\left(x_{1}+k\left(x_{2}-x_{1}\right), y_{1}+k\left(y_{2}-y_{1}\right)\right), k \in \mathbf{Z}$