So I'm given that R is an equivalence relation on Z, and that adding classes in the natural way is well defined. That is, class(a)+class(b) = class(a + b). I want to show that R can only be equality mod n (for some n) or actual equality. I'm provided a hint to take into account the smallest difference present between elements of the same equivalence class...I'm pretty stumped unfortunately.
I think I want to find such a minimal difference between elements and show that any other difference between elements in that class is a multiple of the minimal. The Euclidean algorithm seems relevant to that approach, but methinks I might be entirely misguided...