# 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

## 1 Answer

Yes, we call them congruence/residue classes. Basically, all the numbers that leave remainder 0 when divided by 3 are in one congruence class. Numbers that leave remainder 1 and 2 have their own classes.

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 $14 \equiv 2 \mod(3)$.

I sincerely hope this was something you just came up with, because you've just stumbled upon the foundations of the very exciting world of abstract algebra and number theory. Search for topics like modular arithmetic and/or congruence/residue classes. Enjoy!

• "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
• @anon Thank you for pointing that out, there should be no errors now, I've edited and am now going to sleep. – naslundx Feb 26 '14 at 23:49
• 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 $[0]_2$ and oddness means the number belongs to residue class $[1]_2$. The classifications I was clumsily calling 3-"parity" would be residue classes $[0]_3$, $[1]_3$ and $[2]_3$. Does that seem right? – Pat Muchmore Feb 27 '14 at 0:47
• @PatMuchmore Yes that's correct. – anon Feb 27 '14 at 1:00