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 [53], [73], [93], and [113]
Under addition we the have -[53]=[-53], and so on. Multiplication is a bit trickier. Use the Euclidean algorithm to find a,b such that 53a+365b=1. Then $\displaystyle [53]^{-1}=[a]$. However keep in mind that [73] is NOT an element of the multiplicative group Z/365Z (because (73,365)=73).