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: .

(b) exactly one lattice point?

. . .Yes . . . Consider the line: .

. . .It passes through the Origin and *no** *__other__ lattice point.

(c) exactly two lattice points?

. . .No.

If a line passes through two lattice points: .

. . 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.