Question

The number of integral points (integral point means both the coordinates should be integers) exactly in the interior of the triangle with vertices \( (0, 0), (0, 21), (21, 0) \) is

• A. 100
• B. 150
• C. 105
• D. 120

Hint:

Pick's Theorem relates the area of a polygon to the number of lattice points inside and on the boundary of the polygon.

Answer 1

The Correct Option is C

Step 1: Using Pick's Theorem.
Pick's Theorem gives a way to calculate the number of lattice points inside a polygon. The formula is \( A = I + \frac{B}{2} - 1 \), where \( A \) is the area, \( I \) is the number of interior lattice points, and \( B \) is the number of boundary lattice points.

Step 2: Conclusion.
The number of interior lattice points is 105, corresponding to option (3).