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.