1. lattice points

Is it possible for a line to go through
(a) no lattice points? No
(b) exactly one lattice point? No
(c) exactly two lattice points? No

2. Hello, Veronica1999!

Is it possible for a line to go through

(a) no lattice points? No

(b) exactly one lattice point? No

(c) exactly two lattice points? No

(a) no lattice points?
. . .Yes . . . Consider the horizontal line: .$\displaystyle y \,=\,\frac{1}{2}$

(b) exactly one lattice point?
. . .Yes . . . Consider the line: .$\displaystyle y \,=\,\sqrt{2}\,x$

. . .It passes through the Origin and no other lattice point.

(c) exactly two lattice points?
. . .No.

If a line passes through two lattice points: .$\displaystyle P_1(x_1,y_1)\,\text{ and }\,P_2(x_2,y_2)$
. . it will pass through infinitely many more lattice points.

Code:
                                   (x3,y3)
o
*   :
*       : y2-y1
(x2,y2)*           :
o - - - - - - - +
*   :      x2-x1
*       :y2-y1
(x1,y1)*           :
o - - - - - - - +
x2-x1

I'll let you work through the reasoning.