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# Def/Residue

Definition of Residue: Suppose that $n$ is a positive integer. A residue, mod $n$, is an equivalence class of integers, with respect to the equivalence relation of congruence mod $n$.

For example, there are two residues, mod $2$. One residue, denoted $\bar 0$ is the set of even integers, and the other residue, denoted $\bar 1$, is the set of odd integers.

For every positive integer $n$, there are $n$ residues, mod $n$. If $a$ is an integer, and $n$ is understood from context, we write $\bar a$ for the residue (mod $n$) containing $a$. For example, when $n = 7$, $\bar 3$ denotes the set: $$\bar 3 = \{ \ldots, -11, -4, 3,0,7, \ldots \}.$$ In this way, we see that $\bar 3 = \overline{10}$ (when working with residues mod $7$).

Modular arithmetic is a term used to describe the arithmetic (e.g., addition, subtraction, and multiplication) of residues.

## Logical Connections

This definition logically relies on the following definitions and statements: Def/Equivalence class, Def/Integer, Def/Equivalence relation, Def/Congruent, Def/Even, Def/Odd

The following statements and definitions logically rely on the material of this page: Def/Invertible residue, Def/Legendre symbol, Def/Quadratic residue, Def/Reduce mod n, State/Arithmetic of residues is well-defined, State/Eulers criterion, State/Fermat Euler theorem, State/Fermats little theorem, State/Multiplicative inverses exist mod p, State/Relative primality to the modulus is equivalent to invertibility of a residue, State/There are no zero divisors mod p, and State/Zolotarevs lemma

To visualize the logical connections between this definition and other items of mathematical knowledge, you can visit any of the following clusters, and click the "Visualize" tab: Clust/Modular arithmetic