2020 AIME II Problem 12

Let m and n be odd integers greater than 1. An m\times n rectangle is made up of unit squares where the squares in the top row are numbered left to right with the integers 1 through n, those in the second row are numbered left to right with the integers n + 1 through 2n, and so on. Square 200 is in the top row, and square 2000 is in the bottom row. Find the number of ordered pairs (m,n) of odd integers greater than 1 with the property that, in the m\times n rectangle, the line through the centers of squares 200 and 2000 intersects the interior of square 1099.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.