Points with integer coordinates

Say there was a point with integer coordinates, $(a,b)$ , such that the point lies on the line. The slope of the line connecting $(a,b)$ with $(0,0)$ must be the same as the line connecting $(6000,5052)$ with $(0,0)$ , or $\frac{421}{500}$

$=> \frac{b}{a} = \frac{421}{500}$

$=> b = \frac{421\cdot a}{500}$

Thus, if $b$ is an integer, $a$ must be a multiple of $500$ , from $1$ to $6000$ . *Note - $0$ doesn’t work, as $0$ isn’t positive.

This means that the only coordinates possible are:

• $(500,421)$
• $(1000,842)$
• $(1500,1263)$
• $(2000,1684)$
• $(2500,2105)$
• $(3000,2526)$
• $(3500,2947)$
• $(4000,3368)$
• $(4500,3789)$
• $(5000,4210)$
• $(5500,4631)$
• $(6000,5052)$

This makes a total of $\fbox{12}$ points