Thread: 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

Are my answers correct?

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.