The unit squares on the coordinate plane that have four lattice point vertices are colored black orwhite, as on a chessboard, shown on the diagram below. For an ordered pair $(m, n)$, let $OXZY$ be the rectangle with vertices $O = (0, 0)$, $X = (m, 0)$, $Z = (m, n)$ and $Y = (0, n)$. How many ordered pairs $(m, n)$ of nonzero integers exist such that rectangle $OXZY$ contains exactly 32 black squares?