Question 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 $1099.$