equation solving - How to find lattice points on a line segment?

How do I find points on the line segment joining {-4, 11} and {16, -1} whose coordinates are positive integers?

Answer

Is this what you are searching for?

a = {-4, 11};
b = {16, -1};

dy = (b[[2]] - a[[2]])/(b[[1]] - a[[1]]);


offset = u /. Solve[a[[2]] == dy*a[[1]] + u, u][[1]];

coords = {x, 
   y} /. {Reduce[y == dy*x + offset && x > 0 && y > 0, {x, y}, 
     Integers] // ToRules}

(* {{1, 8}, {6, 5}, {11, 2}} *)

Graphics[{PointSize[Large], Point[{a, b}], Red, Point[coords], 
  Line[{a, b}]}, Axes -> True, GridLines -> {Range[16], Range[16]}, 
 ImageSize -> 640]

Mathematica graphics

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