Solving each the equations, we find that either $x = 108$, $x = 60$, $x = 45$, or $x = 72$.

The exterior angle of a regular polygon must be factor of 360, so this rules out $x = 108$.

For the remaining three cases, $60$º is the exterior angle for regular hexagons, $45$º is for regular octagons, and $72$º is for regular pentagons.

Only the pentagon has a prime number of sides. Therefore, the polygon has 5 sides.