The gravitational potential energy of a $200\,\text{kg}$ satellite $3000\,\text{km}$ above the surface of the Earth is $U=-\frac{GMm}{r}=-8.5 \times 10^9\,\text{J}$.
Would the gravitational potential energy of this satellite be greater or less if it was at an altitude of $3000 \,\text{km}$ above the Moon? Explain.
The answers say that the gravitational force of attraction on the Moon is less than that of on Earth, so the potential energy is less.
I am struggling to understand this: $g=\frac{GM}{r^2}$ is a smaller number on the Moon.
If we write $U=-\frac{GMm}{r}=-\frac{GM}{r^2}\times mr$
From this we see the quantity $\frac{GM}{r^2} $ is a smaller number on the Moon, and $r$ is a smaller number on the Moon (radius of Moon is less than that of Earth). So, $U$ should be a smaller negative number, meaning $U$ is greater on Moon than that on Earth.
This goes against the answers and against the intuition that indeed if force of gravity is less than you have less gravitational potential energy.
How can we mathematically explain this situation?