elementary number theory - How to find primitive roots modulo products of primes and other composites?

I know how to find the primitive roots modulo $23$ and and the primitive roots modulo $23^2=529$, in which we are finding the primitive roots of prime powers.

My questions are what if we want to find the primitive roots of $46$ $(=2\times23)$ and $12167$ $(=23\times529)$?

How can we relate to the primitive roots of $23$ and $529$ that we had found previously? Which theorems can we use?

Many thanks for the helps!