How Many Points between two points?

HINT: Let one of the points be $\langle a,b\rangle$ and the other $\langle c,d\rangle$; then the number of lattice points on the line segment joining them is the same as the number on the line segment joining $\langle 0,0\rangle$ to $\langle c-a,d-b\rangle$. Thus, you might as well focus on counting the number of lattice points on the segment joining the origin to $\langle m,n\rangle$ for integers $m$ and $n$. Look at the equation of the line containing this segment: it’s

$$y=\frac{n}mx\;.$$

Suppose that when you reduce $\frac{n}m$ to lowest terms, you get $\frac{q}r$. Then your equation is

$$y=\frac{q}rx\;,$$

and $y$ is an integer if and only if $r\mid x$.

Added: Suppose that the points are $\langle -2,55\rangle$ and $\langle 1011,1055\rangle$. I’d look at the segment from the origin to $\langle 1011-(-2),1055-55\rangle=\langle 1013,1000\rangle$. It lies on the line

$$y=\frac{1000}{1013}x\;.$$

Any lattice point on that line must have both $x$ and $y$ integers, so suppose that $x$ is an integer. When is $\frac{1000}{1013}x$ an integer? The fraction is in lowest terms, so this occurs only when $x$ is a multiple of $1013$. On the other hand, we’re looking only at the segment between $\langle 0,0$ and $\langle 1013,1000\rangle$, so clearly we must have $0\le x\le 1013$. How many multiples of $1013$ are there in this range? Just two, $0$ and $1013$. Thus, the endpoints are the only lattice points on that segment. Translating it parallel to itself up and to the left by adding $\langle 2,-55\rangle$ to restore the original endpoints doesn’t change the number of lattice points, so $\langle -2,55\rangle$ and $\langle 1011,1055\rangle$ are the only lattice points on the original segment.