Just wondering if there was a method to find inverses besides just guess and check. I need to find for Z/365Z the inverses of , , , and 
Under addition we the have -=[-53], and so on. Multiplication is a bit trickier. Use the Euclidean algorithm to find a,b such that 53a+365b=1. Then . However keep in mind that  is NOT an element of the multiplicative group Z/365Z (because (73,365)=73).