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 $\mathrm{mod}\ n$?
 A: 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.
A: 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 $6$.

A: Simply say "congruent to $a$ modulo $m$" to read "$\equiv a \pmod{m}$".
A: 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 $N=3q+1$.
