1. Identify the triangle

The triangle has vertices: $A(0,0),\quad B(0,41),\quad C(41,0).$

Its sides are:

• $x=0$
• $y=0$
• the line through $(0,41)$ and $(41,0)$, which is $x+y=41.$

So the interior points with integer coordinates are the lattice points satisfying $x>0,\quad y>0,\quad x+y<41.$

2. Count integer solutions

We need the number of positive integer solutions of $x+y<41.$

For each integer $x=1,2,3,\dots,39$, the possible values of $y$ are $1,2,3,\dots,40-x,$ because $x+y\le 40$.

So for each $x$, the number of choices for $y$ is $40-x.$

Hence total number of interior lattice points is $\sum_{x=1}^{39}(40-x).$

Now, $\sum_{x=1}^{39}(40-x)=39+38+\cdots+1.$

This is the sum of first $39$ natural numbers: $\frac{39\cdot 40}{2}=780.$

3. Alternative check using Pick's Theorem

Pick's theorem says: $\text{Area}=I+\frac{B}{2}-1,$ where $I$ is the number of interior lattice points and $B$ is the number of boundary lattice points.

• Area of triangle: $\frac{1}{2}\cdot 41\cdot 41=\frac{1681}{2}.$

• Boundary points:

  • on $x$-axis from $(0,0)$ to $(41,0)$: $42$
  • on $y$-axis from $(0,0)$ to $(0,41)$: $42$
  • on line segment from $(0,41)$ to $(41,0)$: $\gcd(41,41)+1=42$

Adding these and subtracting the 3 vertices counted twice: $B=42+42+42-3=123.$

Then