How many points with integer coordinates lie strictly inside the triangle with vertices (0, 0), (12, 0) and (0, 8)?

Use Pick's theorem: Area = I + B/2 − 1, where I = interior lattice points and B = boundary lattice points.

Area = ½ × 12 × 8 = 48.

Boundary points: along x-axis side 12 steps, along y-axis side 8 steps, along the hypotenuse gcd(12, 8) = 4 steps → B = 12 + 8 + 4 = 24.

I = 48 − 24/2 + 1 = 37.

(Direct count: for x = 1 to 11, count y ≥ 1 with 8x + 12y < 96, i.e. 2x + 3y < 24 — this also gives 37.)