How can a synchronous motor have RPM other than 3000/3600?
this motor has a reduction gear that slows it down, to 5 rpm at 50Hz or 6rpm at 60 Hz.
Actually the rotation speed of a synchronous motor in rpm is given by:
$$\frac{2 \times 60 \times f} n$$
where
f = frequency (Hz)
n = number of poles.
The motor in your photo cannot have 1200 poles, so it surely has a stepdown gear.
It is said, that synchronous motor is rotating at the same speed as magnetic field rotates. But three-phase made magnetic field rotates as the frequency of current, i.e. 50 or 60 times per second, which means 3000 or 3600 RPM
Incorrect in your last sentence. A synchronous motor speed has the following formula: -
What you were failing to consider is the number of poles inside the motor.
EDIT - What is 120 here?
Consider this 3-phase machine: -
There are six windings because one phase has to have two poles and poles are counted in pairs so the "2" in the "2x60=120" is for that.
The 60 converts revs per second into revs per minute.