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Math Help - lattice points

  1. #1
    Member Veronica1999's Avatar
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    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?
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  2. #2
    Super Member

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    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: . y \,=\,\frac{1}{2}


    (b) exactly one lattice point?
    . . .Yes . . . Consider the line: . 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: . 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.

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