Count all lattice points

Call the triangle formed in the question $\Delta ABC$ , where point $B$ 's coordinate is $(8, 0)$ , point $A$ 's coordinate is $(0, 0)$ and point $C$ 's coordinate is $(0, 7)$ .

Consider a line that satisfies $y=-x+8$ . This line and the x-y axis forms a triangle called $\Delta ABD$ , where point $D$ 's coordinate is $(0, 8)$ . Triangle $\Delta ABD$ contains $7+6+...+1 = 28$ numbers of lattice points.

Now consider the case where $x=0$ , on the triangle $\Delta ABD$ , lattice point $(0, 7)$ is included, but on the triangle $\Delta ABC$ , this point is not. By induction, in total, 7 lattice points were excluded by triangle $\Delta ABC$ , which makes the solution $28-7=21$ .