Is there an equivalent or an extension of the concept of parity for divisibility by numbers other than 2?

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?

• Try looking for "modular arithmetic". – David Feb 26 '14 at 23:19
• Keywords: modular arithmetic, integers mod n, congruence, residue class. This is just the tip of the iceberg leading into the basic material of elementary number theory and abstract algebra. – anon Feb 26 '14 at 23:22

There are some very interesting properties associated with them. For example, adding or multiplying two numbers, the sum has a remainder which is the sum or product of the remainders. For example $6$ has remainder $0$ when divided by $3$ and $8$ has remainder $2$. Hence, $6+8=14$ has remainder $0+2=2$ when divided by $3$. Mathematicians write this as $14 \equiv 2 \mod(3)$.
• "if you add two numbers within the same class, you end up with another number in the same class" This is incorrect. For instance, $38$ divided by $8$ leaves a remainder of $6$, not $7$. The truth is even more interesting: the classes which $a+b$ and $ab$ are in are determined by the classes $a$ and $b$ were in (thus, the relation of being in the same class is more than just a typical relation: it is a congruence relation), and in this way the classes themselves form a number system (the official term being "ring"). – anon Feb 26 '14 at 23:38
• Thank you, I think "residue class" is precisely the terminology I was looking for. If I'm understanding correctly, it looks like one could say that in standard parity "evenness" means the number belongs to residue class $_2$ and oddness means the number belongs to residue class $_2$. The classifications I was clumsily calling 3-"parity" would be residue classes $_3$, $_3$ and $_3$. Does that seem right? – Pat Muchmore Feb 27 '14 at 0:47