Let $ABCD$ be a rectangle and let $\overline{DM}$ be a segment perpendicular to the plane of $ABCD$. Suppose $DM$ has integer length, and the lengths $MA$, $MC$, $MB$ are positive integers that increase by $3$ at each step (in this order). What is the volume of pyramid $MABCD$?

1. Since $DM\perp$ plane $ABCD$ and $ABCD$ is a rectangle, $MA^2=DM^2+DA^2$, $MC^2=DM^2+DC^2$, and $MB^2=DM^2+DA^2+DC^2$, so $DM^2=MA^2+MC^2-MB^2$.
2. With $MA=n$, $MC=n+3$, $MB=n+6$: $DM^2=n^2-6n-27$, i.e. $(n-3)^2-DM^2=36$, so $(n-3-DM)(n-3+DM)=36$.
3. Both factors have the same parity, hence are both even: $(2,18)$ or $(6,6)$. The pair $(6,6)$ gives $DM=0$, so $n-3=10$, $n=13$ and $DM=8$.
4. Then $DA^2=13^2-8^2=105$ and $DC^2=16^2-8^2=192$, so $[ABCD]=\sqrt{105\cdot192}=24\sqrt{35}$.
5. The volume is $\frac13\cdot24\sqrt{35}\cdot8=\boxed{64\sqrt{35}}$.