The problem is these orbital periods are generally not rational multiples of each other, so their GCD is undefined (or 0, in a sense). Some exceptions to this

The problem is these orbital periods are generally not rational multiples of each other, so their GCD is undefined (or 0, in a sense). Some exceptions to this exist due to orbital resonance: Pluto and Neptune are in 2:3 resonance, and in the Jovian system Ganymede, Europa, and Io are in 1:2:4 resonance.