Modular Arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book '' Disquisitiones Arithmeticae'', published in 1801.

A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Simple addition would result in , but clocks "wrap around" every 12 hours. Because the hour number starts over at zero when it reaches 12, this is arithmetic ''modulo'' 12. In terms of the definition below, 15 is ''congruent'' to 3 modulo 12, so "15:00" on a 24-hour clock is displayed "3:00" on a 12-hour clock.

Congruence

Given an integer , called a modulus, two integers and are said to be congruent modulo , if is a divisor of their difference (that is, if there is an integer such that ).

Congruence modulo is a congruence relation, meaning that it is an equivalence relation that is compatible with the operations of addition, subtraction, and multiplication. Congruence modulo is denoted:

:$a \equiv b \pmod n.$

The parentheses mean that applies to the entire equation, not just to the right-hand side (here, ). This notation is not to be confused with the notation (without parentheses), which refers to the modulo operation. Indeed, denotes the unique integer such that and $a \equiv b \; (\text\; n)$ (that is, the remainder of $b$ when divided by $n$).

The congruence relation may be rewritten as
:$a = kn + b,$
explicitly showing its relationship with Euclidean division. However, the here need not be the remainder of the division of by Instead, what the statement asserts is that and have the same remainder when divided by . That is,
:$a = pn + r,$
:$b = qn + r,$
where is the common remainder. Subtracting these two expressions, we recover the previous relation:
:$a - b = kn,$
by setting

Examples

In modulus 12, one can assert that:

:$38 \equiv 14 \pmod$

because , which is a multiple of 12. Another way to express this is to say that both 38 and 14 have the same remainder 2, when divided by 12.

The definition of congruence also applies to negative values. For example:

:$\begin 2 &\equiv -3 \pmod 5\\ -8 &\equiv 7 \pmod 5\\ -3 &\equiv -8 \pmod 5. \end$

Properties

The congruence relation satisfies all the conditions of an equivalence relation:
* Reflexivity:
* Symmetry: if for all , , and .
* Transitivity: If and , then

If and or if then:
* for any integer (compatibility with translation)
* for any integer (compatibility with scaling)
* for any integer
* (compatibility with addition)
* (compatibility with subtraction)
* (compatibility with multiplication)
* for any non-negative integer (compatibility with exponentiation)
* , for any polynomial with integer coefficients (compatibility with polynomial evaluation)

If , then it is generally false that . However, the following is true:
* If where is Euler's totient function, then —provided that is coprime with .

For cancellation of common terms, we have the following rules:

* If , where is any integer, then
* If and is coprime with , then
* If and , then

The modular multiplicative inverse is defined by the following rules:

* Existence: there exists an integer denoted such that if and only if is coprime with . This integer is called a ''modular multiplicative inverse'' of modulo .
* If and exists, then (compatibility with multiplicative inverse, and, if , uniqueness modulo )
* If and is coprime to , then the solution to this linear congruence is given by

The multiplicative inverse may be efficiently computed by solving Bézout's equation $ax + ny = 1$ for $x,y$—using the Extended Euclidean algorithm.

In particular, if is a prime number, then is coprime with for every such that ; thus a multiplicative inverse exists for all that is not congruent to zero modulo .

Some of the more advanced properties of congruence relations are the following:
* Fermat's little theorem: If is prime and does not divide , then .
* Euler's theorem: If and are coprime, then , where is Euler's totient function
* A simple consequence of Fermat's little theorem is that if is prime, then is the multiplicative inverse of . More generally, from Euler's theorem, if and are coprime, then .
* Another simple consequence is that if where is Euler's totient function, then provided is coprime with .
* Wilson's theorem: is prime if and only if .
* Chinese remainder theorem: For any , and coprime , , there exists a unique such that and . In fact, where is the inverse of modulo and is the inverse of modulo .
* Lagrange's theorem: The congruence , where is prime, and is a polynomial with integer coefficients such that , has at most roots.
* Primitive root modulo : A number is a primitive root modulo if, for every integer coprime to , there is an integer such that . A primitive root modulo exists if and only if is equal to or , where is an odd prime number and is a positive integer. If a primitive root modulo exists, then there are exactly such primitive roots, where is the Euler's totient function.
* Quadratic residue: An integer is a quadratic residue modulo , if there exists an integer such that . Euler's criterion asserts that, if is an odd prime, and is not a multiple of , then is a quadratic residue modulo if and only if
::$a^ \equiv 1 \pmod p.$

Congruence classes

Like any congruence relation, congruence modulo is an equivalence relation, and the equivalence class of the integer , denoted by , is the set . This set, consisting of all the integers congruent to  modulo , is called the congruence class, residue class, or simply residue of the integer modulo . When the modulus is known from the context, that residue may also be denoted .

Residue systems

Each residue class modulo may be represented by any one of its members, although we usually represent each residue class by the smallest nonnegative integer which belongs to that class (since this is the proper remainder which results from division). Any two members of different residue classes modulo are incongruent modulo . Furthermore, every integer belongs to one and only one residue class modulo .

The set of integers is called the least residue system modulo . Any set of integers, no two of which are congruent modulo , is called a complete residue system modulo .

The least residue system is a complete residue system, and a complete residue system is simply a set containing precisely one representative of each residue class modulo . For example. the least residue system modulo 4 is . Some other complete residue systems modulo 4 include:

*
*
*
*
*
*

Some sets which are ''not'' complete residue systems modulo 4 are:

*, since 6 is congruent to 22 modulo 4.
*, since a complete residue system modulo 4 must have exactly 4 incongruent residue classes.

Reduced residue systems

Given the Euler's totient function , any set of integers that are relatively prime to and mutually incongruent under modulus is called a reduced residue system modulo . The set from above, for example, is an instance of a reduced residue system modulo 4.

Integers modulo ''n''

The set of all congruence classes of the integers for a modulus is called the ring of integers modulo , and is denoted $\mathbb/n\mathbb$, $\mathbb/n$, or $\mathbb_n$. The notation $\mathbb_n$ is, however, not recommended because it can be confused with the set of -adic integers. The ring $\mathbb/n\mathbb$ is fundamental to various branches of mathematics (see below).

The set is defined for ''n'' > 0 as:

:$\mathbb/n\mathbb = \left\ = \left\.$

(When , $\mathbb/n\mathbb$ is not an empty set; rather, it is isomorphic to $\mathbb$, since .)

We define addition, subtraction, and multiplication on $\mathbb/n\mathbb$ by the following rules:

* $\overline_n + \overline_n = \overline_n$
* $\overline_n - \overline_n = \overline_n$
* $\overline_n \overline_n = \overline_n.$

The verification that this is a proper definition uses the properties given before.

In this way, $\mathbb/n\mathbb$ becomes a commutative ring. For example, in the ring $\mathbb/24\mathbb$, we have
:$\overline_ + \overline_ = \overline_= \overline_$
as in the arithmetic for the 24-hour clock.

We use the notation $\mathbb/n\mathbb$ because this is the quotient ring of $\mathbb$ by the ideal $n\mathbb$, a set containing all integers divisible by , where $0\mathbb$ is the singleton set . Thus $\mathbb/n\mathbb$ is a field when $n\mathbb$ is a maximal ideal (i.e., when is prime).

This can also be constructed from the group $\mathbb Z$ under the addition operation alone. The residue class is the group coset of in the quotient group