Gotta say, it's been about two weeks working on this one, burning my brain. I took a lot of wrong turns and learned a lot about modular arithmetic and number theory.

To anyone stumped, Dario Alpern's site is indispensable. The calculator page and the method page explain much of what you need to implement. Then, also brushing up and understanding how to solve quadratic congruences. And definitely build yourself some testing suites to make sure you are getting the right answers! Good luck!