Modulo (jargon)

In the mathematical branch of number theory, the residue class of a number is a number modulo the set of all numbers that have the same remainder when divided by such.

Definition

It is an integer other than 0 and is an arbitrary integer. The residue class of modulo written,

Is the equivalence class of with respect to the congruence modulo, ie the set of integers that when divided by the same remainder as result. It thus consists of all integers that result from the addition of integer multiples of:

An element of a coset is also referred to as the representative of the residue class. Often one uses the standard representatives.

The set of all residue classes modulo one writes frequently than or. It has the structure and elements of a ring and is therefore called the residue class ring. Just when is a prime number, there is even the structure of a finite field.

A residue class modulo is called residue class if their elements are relatively prime. (. If this applies to an element, then also for all others) The amount of the reduced residue classes is the group of units (or) in the residue class ring; it is called prime residue class group and includes the multiplicative invertible residue classes.

Examples

• The residual of 0 modulo 2, the set of even numbers.
• The residual of 1 modulo 2, the set of odd numbers.
• The residual modulus of 0, the amount of a multiple of.
• The residue class of 1 modulo 3, the amount

Generalization

Is a ring and an ideal, so hot sets of the form

Residue classes modulo. Is commutative, or is a two-sided ideal, then the set of residue classes modulo has a natural ring structure and is called the residue class ring, quotient ring or factor ring modulo. is represented by elements, wherein the residue classes and, if applicable in the same.

• Number Theory
• Ring theory