Question

Consider a triangle drawn on the X-Y plane with its three vertices at $(41, 0)$, $(0, 41)$ and $(0, 0)$, each vertex being represented by its $(X, Y)$ coordinates. The number of points with integer coordinates inside the triangle (excluding all the points on the boundary) is

Accepted Answer

A. The interior of the triangle is bounded by $x > 0$, $y > 0$, and $x + y < 41$. Let $x + y = k$. For integer solutions inside the triangle, $k$ can range from $2$ to $40$. For a given $k$, the number of positive integer solutions $(x, y)$ is $k - 1$. Total interior points $= \sum_{k=2}^{40} (k - 1) = 1 + 2 + 3 + \dots + 39 = \frac{39 	imes 40}{2} = 780$.

Interior integer points of a triangle

Explanation

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