A number is even if it leaves no remainder when divided by 2 and odd if there is a remainder. The same basic idea could be extended to whether or not a number is divisible by, for example, 3 without remainder, and we could classify numbers by this 3-"parity". What's potentially interesting about that to me is that it would generate three possibilities rather than 2, since some numbers would leave no remainder, some numbers would leave a remainder of 1 and some numbers would leave a remainder of 2. For example, in some sequences with a particularly ternary structure, it seems like numbers might behave similarly within their 3-"parity" class, but that the three classes would behave differently from one another.
Google search and JSTOR search so far have been fruitless for me, but I assume that's because I lack the proper vocabulary for this classification. Obviously, "parity" is a pretty poor term for something with three or more classifications. What, if anything, is the proper terminology for this classification, and where can I learn more?