Why the spurs of a signal after a divider have the same offset as before the divider?

Say there are spurs 1 MHz from the carrier, on either side, that are -30 dBc. Think of them as phase modulation sidebands, with a modulation frequency of 1 MHz. If you divide by 2 the carrier frequency halves, the modulation frequency of 1 MHz does not change, but the phase deviation does (gets smaller), so the spur locations in frequency from the carrier DO NOT change, but the amplitude of the spurs drop.

Thank you very much!
Then how much does the amplitude of the spurs drop? 6dBc?
Are there some books or paper talks about this? Just one is ok!

If you divide a signal by 2 the phase angle deviation halves. If you multiply a signal by 2, the phase angle doubles.

divide by 2 is like a 20 log 1/2 = -6 dB

Divide by 10 is like a 20 log 1/10 = -20 dB

Multiply by 2 is like a 20 log 2 = +6 dB

and so on...