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# Lattice points

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A lattice point is an ordered pair (x, y) where both x and y are integers. A triangle is formed
by the three points (1, 1), (9, 1), and (9, n). For what integer value of n > 0 are there exactly 560 lattice
points strictly in the interior of the triangle?

(Yes I know I posted this one a few days ago, but the only answer I got was from a guest who didn't show their work and didn't get the correct answer... so for the people answering this one, please show your work.)

Nov 2, 2022

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Wait, but how did you get it? Can you show how you got your answer?

Chandalier  Nov 2, 2022
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Complete the rectangle.

Discounting the boundary, there will be 7 verticals and n - 2 horizontals producing a total of 7(n - 2) lattice points.

Assuming for the moment that there are no lattice points on the diagonal, we need 7( n -2) = 2×560, so n = 162.

Check now that with n = 162 there are no lattice points on the diagonal meaning that there will be 560 lattice points within each of the two triangles.

Nov 6, 2022