Solving simultaneous multiple chinese remainder theorem on the quantum computer - possible?
If you prepare a superposition containing the various input problems, a quantum computer will allow you to compute a superposition containing the various outputs.

That being said, I'm not sure why you would want to do this. You only get to sample one of the outputs, and you don't get to pick which one. You'll get one of them at random. Presumably you would instead do some quantum computation on the output superposition, perhaps use a QFT to sample from their frequency space... but just remember that this means you must erase the input! Otherwise you won't be able to interfere the various output cases.