# Residual class

In mathematical branch of number theory that is residual of a number modulo a number of the set of numbers that at Division by leave the same remainder as . ${\ displaystyle a}$ ${\ displaystyle m}$${\ displaystyle m}$${\ displaystyle a}$

## definition

Let it be an integer other than 0 and any integer. The remainder class of modulo , written ${\ displaystyle m}$${\ displaystyle a}$${\ displaystyle a}$${\ displaystyle m}$

${\ displaystyle a + m \ mathbb {Z},}$

is the equivalence class of modulo with respect to congruence, i.e. the set of integers that result when dividing by the same remainder as . It therefore consists of all integers that result from the addition of integer multiples of : ${\ displaystyle a}$${\ displaystyle m}$${\ displaystyle m}$${\ displaystyle a}$${\ displaystyle b}$${\ displaystyle a}$${\ displaystyle m}$

${\ displaystyle a + m \ mathbb {Z} = \ {b \ mid b = a + km \ \, \ mathrm {f {\ ddot {u}} r \ ein} \ \, k \ in \ mathbb {Z } \} = \ {b \ mid b \ equiv a \; ({\ rm {mod}} \; m) \}}$.

An element of a remainder class is also called a representative of the remainder class. The standard representatives are often used . ${\ displaystyle 0,1,2, \ dots, m-1}$

The set of all remainder classes modulo is often written as or . It has the elements and the structure of a ring and is therefore called a residual class ring . Exactly when it is a prime number even the structure of a finite field results . ${\ displaystyle m}$${\ displaystyle \ mathbb {Z} / m \ mathbb {Z}}$${\ displaystyle \ mathbb {Z} _ {m}}$${\ displaystyle m}$${\ displaystyle m}$

A residue class modulo is residue class if their elements are relatively prime to have. (If this applies to one element, then also to all others.) The set of prime remainder classes is the group of units (or ) in the remainder class ring ; it is called the prime remainder class group and includes the multiplicatively invertible remainder classes. ${\ displaystyle m}$${\ displaystyle m}$ ${\ displaystyle (\ mathbb {Z} / m \ mathbb {Z}) ^ {\ times}}$${\ displaystyle \ mathbb {Z} _ {m} ^ {*}}$ ${\ displaystyle \ mathbb {Z} / m \ mathbb {Z}}$

## Examples

• The remainder class of 0 modulo 2 is the set of even numbers.
• The remainder class of 1 modulo 2 is the set of odd numbers.
• The remainder class of 0 modulo is the set of multiples of .${\ displaystyle m}$${\ displaystyle m}$
• The remainder class of 1 modulo 3 is the quantity ${\ displaystyle \ {\ ldots, -8, -5, -2,1,4,7,10, \ ldots \}.}$

## generalization

If there is a ring and an ideal , then quantities of form are called ${\ displaystyle A}$${\ displaystyle I \ subseteq A}$

${\ displaystyle a + I = \ {a + i \ mid i \ in I \}}$

Remaining classes modulo . Is commutative , or is a two-sided ideal, then the set of remainder classes modulo has a natural ring structure and is called a remainder class ring , quotient ring or factor ring modulo . is represented by elements in , where the remainder classes and in are the same if applicable. ${\ displaystyle I}$${\ displaystyle A}$ ${\ displaystyle I}$${\ displaystyle A / I}$${\ displaystyle I}$${\ displaystyle I}$${\ displaystyle A / I}$${\ displaystyle A}$${\ displaystyle a + I}$${\ displaystyle b + I}$${\ displaystyle A / I}$${\ displaystyle from \ in I}$