Problem:

Let where and .

When is periodic?

My Attempt:

Let be periodic.

Let the time period of , and be , and respectively.

Since has period , .

So and by the same logic .

So

.

Not really sure that this gets me anywhere...

As far as I am aware in order for to be periodic then the time periods of and must have a rational LCM, so is periodic if there exist and such that . Is this correct? If so how can I show this?

Thanks in advance for your help!