# How is this property called for mod?

We have a name for the property of integers to be $$0$$ or $$1$$ $$\mathrm{mod}\ 2$$ - parity.

Is there any similar name for the remainder for any other base? Like a generalization of parity? Could I use parity in a broader sense, just to name the remainder $$\mathrm{mod}\ n$$?

• How about congruence? – Umberto P. Feb 25 '19 at 11:37
• So should I say (variable)'s congruence class or value? – dEmigOd Feb 25 '19 at 11:43
• I had the same question and thought about terminology for $n = 3$. A friend of mine came up with a very neat suggestion: flat, short and long. I don't know if there is any other generalization, but I would very much like to hear people talking about remainders mod 3 like this. I guess the reason why this has a name for $n=2$ is that it appears commonly in everyday life as opposed to most of the other numbers. When talking about time, the 12 is dropped at all, but this is somewhat different maybe – kesa Feb 25 '19 at 15:44

## 4 Answers

Simply say "congruent to $$a$$ modulo $$m$$" to read "$$\equiv a \pmod{m}$$".

Actually there is a standard name: residue.

There are $$5$$ residues modulo $$5$$, namely $$0,1,2,3,4$$.

Every prime greater than $$3$$ falls into only $$2$$ residue-classes modulo $$6$$.

In a broader sense, when you are dealing with congruences you are dealing with an equivalence relation and its equivalence classes.

The basic idea of an equivalence relation is to collect elements that are different but behave in the same manner with respect to some property of interest.

In modular arithmetic this property is having the same rest when divided by a prescribed integer

If $$a=b\bmod m$$ or $$a\equiv_m b$$ you essentially say that $$a$$ and $$b$$ are in the same equivalence class with respect to the equivalence relation $$\equiv_m$$.

So you could say that "$$a$$ and $$b$$ are in the same equivalence class when we look at the remainder upon dividing by $$m$$", which definitely longer and more cumbersome to say "$$a$$ congruent to $$b$$ modulo $$m$$" but nonetheless another way to express the same concept.

Either AlessioDV's good answer, or you could say: "of the form $$nq+r$$". as a representation of $$\equiv r \pmod n$$ for example a number $$N$$ with $$N\equiv 1\pmod 3$$ has property $$N=3q+1$$.