Lattice Points in Math

Let's say we have a grid of regularly spaced lines in $n$-dimensions. Whenever $n$ of these lines intersect at $1$ point, we call the point at which they intersect a lattice point. This terminology will help in higher dimensions, but a simpler way to put it is a point where all of the coordinates of that point are integers. For example, in the 2-dimensional coordinate plane where every point can be given an $x$-coordinate and a $y$-coordinate, if the point has coordinates $(x,y)$ where $x$ and $y$ are integers, then that point is a lattice point. To test your knowledge of lattice points, try these simple questions:

Here is a tougher question from an actual competition! The source is the $2011$ AMC$10$B (American Math Competitions) question $24$

If you solved both of those questions, then you know what a lattice point is!