Lattice point

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A lattice point is a point in a Cartesian coordinate system such that both its $x$- and $y$-coordinates are integers. A lattice point is a point at the intersection of two or more grid lines in a regularly spaced array of points, which is a point lattice. In a plane, point lattices can be constructed having unit cells in the shape of a square, rectangle, hexagon, and other shapes. If not specified, a point lattice is usually a point in a square array. Lattice points are complicated, so don't get stressed if you don't get it right away! Here's an example to help you to understand it better:


A point lattice is constructed by plotting all of the points $(a,b)$ such that $a$ and $b$ are positive integers. How many points in the point lattice lie on the line $y = -3x + 8$?


Notice that $y > 0 \implies -3x + 8 > 0 \implies x \leq 2$. So, $(1,5)$ and $(2,2)$ are the only such points, giving us $\boxed{2}$ points.


See Also