Let $ABCD$ be a rectangle and let $\overline{DM}$ be a segment perpendicular to the plane of $ABCD$. Suppose that $\overline{DM}$ has integer length and the lengths of $\overline{MA},$ $\overline{MC}$, and $\overline{MB}$ are consecutive odd positive integers with a common difference of 4 (in this order). What is the volume of pyramid $MABCD$?