Answer
$7.1\times10^3s$
Work Step by Step
The speed of an object in an orbit of radius $r$ around the Moon is given by $v=\sqrt{\frac{GM_{Moon}}{r}}$ and also $v=\frac{2\pi r}{T}$, where T is the period of the object in orbit. Equate the two expressions and solve for $T$.
$$v=\sqrt{\frac{GM_{Moon}}{r}}=\frac{2\pi r}{T}$$
$$T=2\pi\sqrt{\frac{r^3}{GM_{Moon}}}=2\pi\sqrt{\frac{(R_{Moon}+100km)^3}{GM_{Moon}}}=2\pi\sqrt{\frac{(1.74\times10^6m+1\times10^5m)^3}{(6.67\times10^{-11}Nm^2/kg^2)(7.35\times10^{22}kg)}}$$
$$=7.1\times10^3s$$
