The number of integral points (integral point means both the coordinates should be integer) exactly in the interior of the triangle with vertices (0, 0), (0, 21) and (21, 0) is [2003]

Question

The number of integral points (integral point means both the coordinates should be integer) exactly in the interior of the triangle with vertices (0, 0), (0, 21) and (21, 0) is [2003]

Smart Achievers · Accepted Answer

(2)

Total number of points within the square $O A C B = 20 \times 20 = 400$

$Points on line A B = 20 {(1, 20), (2, 19), (3, 18), \dots, (10, 11), (11, 10), \dots, (20, 1)}$

$∴$ $Points within ∆ O B C and ∆ A B C = 400 - 20 = 380$

$By symmetry, points within ∆ O A B = \frac{380}{2} = 190$

Solution

Q.
The number of integral points (integral point means both the coordinates should be integer) exactly in the interior of the triangle with vertices (0, 0), (0, 21) and (21, 0) is [2003]

1 133

2 190

3 233

4 105

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