What is the apparent rotation time for a planet relative to an other planet?
Note, an observer on Planet A can't see a rotation of Planet B. He can see a the Sun orbiting around him, and Planet B orbiting around the Sun. Thus


If Planet B has an orbit of smaller radius than Planet A, then the observer can see Planet B sometimes before, sometimes after its Sun.
If Planet B has an orbit of larger radius than Planet A, then the observer can see Planet B to cycle on the sky, roughly on the plane of the Ecliptics,  but a little bit faster as the Sun.


The calculation is this:


Planet A takes $\frac{1}{m}$ full circle in a day.
Planet B takes $\frac{1}{n}$ full circle in a day.
Thus, the observer on Planet A can see an $|\frac{1}{m}-\frac{1}{n}|$ full circle of rotation of Planet B in a day.
Thus, he can see a period of Planet B of $\frac{1}{|\frac{1}{m}-\frac{1}{n}|}$ Earth days, which is in simplified form


$$\underline{\underline{|\frac{mn}{m-n}|}}.$$