He further advanced modular arithmetic, greatly simplifying manipulations in number theory.

This remarkably general law allows mathematicians to determine the solvability of any quadratic equation in modular arithmetic.

Gauss also made important contributions to number theory with his 1801 book Disquisitiones Arithmeticae ( Latin, Arithmetical Investigations ), which, among things, introduced the symbol ≡ for congruence and used it in a clean presentation of modular arithmetic, contained the first two proofs of the law of quadratic reciprocity, developed the theories of binary and ternary quadratic forms, stated the class number problem for them, and showed that a regular heptadecagon ( 17-sided polygon ) can be constructed with straightedge and compass.

The original form of the theorem, contained in a third-century AD book The Mathematical Classic of Sun Zi ( 孫子算經 ) by Chinese mathematician Sun Tzu and later generalized with a complete solution called Da yan shu ( 大衍术 ) in a 1247 book by Qin Jiushao, the Shushu Jiuzhang ( 數書九章 Mathematical Treatise in Nine Sections ) is a statement about simultaneous congruences ( see modular arithmetic ).

for all integers which are relatively prime to ( see modular arithmetic ).

These can be of quite general use: for example in modular arithmetic or powering of matrices.

Article 16 of Gauss ' Disquisitiones Arithmeticae is an early modern statement and proof employing modular arithmetic.

* The ring of p-adic integers is the inverse limit of the rings Z / p < sup > n </ sup > Z ( see modular arithmetic ) with the index set being the natural numbers with the usual order, and the morphisms being " take remainder ".

In modular arithmetic, the question of when a linear congruence can be solved is answered by the linear congruence theorem.

In mathematics, modular arithmetic ( sometimes called clock arithmetic ) is a system of arithmetic for integers, where numbers " wrap around " upon reaching a certain value — the modulus.

A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods.

The notion of modular arithmetic is related to that of the remainder in division.

When working with modular arithmetic, each equivalence class is usually represented by its common residue, for example which can be found using long division.

Another example is the construction of finite fields, which proceeds similarly, starting out with the field of integers modulo some prime number as the coefficient ring R ( see modular arithmetic ).

In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic which gives conditions for the solvability of quadratic equations modulo prime numbers.

An ideal can be used to construct a quotient ring similarly to the way that modular arithmetic can be defined from integer arithmetic, and also similarly to the way that, in group theory, a normal subgroup can be used to construct a quotient group.

The integer n is square-free if and only if the factor ring Z / nZ ( see modular arithmetic ) is a product of fields.

That is, for every prime number p with, one has the modular arithmetic relations that either or: the final digits is a 1 or a 5.

Instead of rewriting the current symbol, it can perform a modular arithmetic incrementation on it.

One advantage of this convention is in the use of modular arithmetic as implemented in modern computers.

In modular arithmetic notation, define the function f as follows:

in modular arithmetic, subject to the constraints.

In modular arithmetic, some numbers have a multiplicative inverse with respect to the modulus.

In modular arithmetic, a branch of number theory, a primitive root modulo n is any number g with the property that any number coprime to n is congruent to a power of g modulo n. In other words, g is a generator of the multiplicative group of integers modulo n. That is, for every integer a coprime to n, there is an integer k such that g < sup > k </ sup > ≡ a ( mod n ).

The multiplicative inverse has innumerable applications in algorithms of computer science, particularly those related to number theory, since many such algorithms rely heavily on the theory of modular arithmetic.

# Use the extended Euclidean algorithm to compute k < sup >− 1 </ sup >, the modular multiplicative inverse of k mod 2 < sup > w </ sup >, where w is the number of bits in a word.

where N < sub > 1 </ sub >< sup >− 1 </ sup > denotes the modular multiplicative inverse of N < sub > 1 </ sub > modulo N < sub > 2 </ sub > and vice-versa for N < sub > 2 </ sub >< sup >− 1 </ sup >; the indices k < sub > a </ sub > and n < sub > a </ sub > run from 0 ,..., N < sub > a </ sub >− 1 ( for a

The Euclidean modular multiplicative inverse of 2 mod 3 is 2.

** simplify, use the modular multiplicative inverse if necessary

where is the modular multiplicative inverse of modulo.

# Ensure divides the order of by checking the existence of the following modular multiplicative inverse:,

:: Note that the notation does not denote the modular multiplication of times the modular multiplicative inverse of but rather the quotient of divided by, i. e., the largest integer value to satisfy the relation.

Modular exponentiation can be performed with a negative exponent e by finding the modular multiplicative inverse d of b modulo m using the extended Euclidean algorithm.

In modular arithmetic the set of congruence classes relatively prime to the modulus n form a group under multiplication called the multiplicative group of integers modulo n. It is also called the group of primitive residue classes modulo n. In the theory of rings, a branch of abstract algebra, it is described as the group of units of the ring of integers modulo n. ( Units refers to elements with a multiplicative inverse.

Inversive congruential generators are a type of nonlinear congruential pseudorandom number generator, which use the modular multiplicative inverse ( if it exists ) to generate the next number in a sequence.

The modular inverse k < sup >− 1 </ sup > mod q is the second most expensive part, and it may also be computed before the message hash is known.

* In modular arithmetic, the modular additive inverse of is also defined: it is the number such that.

Image: Modular_pair. svg | In a modular lattice, the maps φ and ψ indicated by the arrows are mutually inverse isomorphisms.

The matrix will have an inverse if and only if its determinant is not zero, and does not have any common factors with the modular base.

Elliptic modular functions can sometimes be expressed as the inverse functions of ratios of hypergeometric functions whose arguments a, b, c are 1, 1 / 2, 1 / 3, ... or 0.

The idea behind our design is modular units, or panelization.

Still another approach to the changeable letter type of sign is a modular unit introduced by Merritt Products, Azusa, Calif..

However, his construction of the Gothic alphabet is based upon an entirely different modular system.

Most games use a standardized and unchanging board ( chess, Go, and backgammon each have such a board ), but many games use a modular board whose component tiles or cards can assume varying layouts from one session to another, or even while the game is played.

The classification of groups of small 2-rank, especially ranks at most 2, makes heavy use of ordinary and modular character theory, which is almost never directly used elsewhere in the classification.

This addition-based modular sum is used in SAE J1708.

He then had little more to publish on the subject ; but the emergence of Hilbert modular forms in the dissertation of a student means his name is further attached to a major area.

Another important feature of the Doom engine is its modular data files, which allow most of the game's content to be replaced by loading custom WAD files.

Because the Eurocard system provided for so many modular card sizes and because connector manufacturers have continued to create new connectors which are compatible with this system, it is a popular mechanical standard which is also used for innumerable " one-off " applications.

The Eurocard standard is also the basis of the " Eurorack " format for modular electronic music synthesizers, popularized by Doepfer and other manufacturers.

In this notation, the use of a vertical bar as delimiter indicates that the argument following it is the " parameter " ( as defined above ), while the backslash indicates that it is the modular angle.

Dualis is a strictly modular partial low-floor car, with all doors in the low-floor sections.

Flying cars fall into one of two styles ; integrated ( all the pieces can be carried in the vehicle ), or modular ( the aeronautical sections are left at the airport when the vehicle is driven ).

* The Aerocar 2000 is a modular design currently in development by Ed Sweeney, owner of one of Moulton Taylor's Aerocars.

This is a modular design, in development.

Freenet is modular and features an API called Freenet Client Protocol ( FCP ) for other programs to use to implement services such as message boards, file sharing, or online chat.

If the field of scalars of the vector space has characteristic p, and if p divides the order of the group, then this is called modular representation theory ; this special case has very different properties.

Windows Server 2008 builds on the technological and security advances first introduced with Windows Vista, and is significantly more modular than its predecessor, Windows Server 2003.

* The ( 3-dimensional ) metaplectic group is a double cover of SL < sub > 2 </ sub >( R ) playing an important role in the theory of modular forms.

This approach is simple and modular, but has the disadvantage that the model itself can be expensive to store, and also that it forces a single model to be used for all data being compressed, and so performs poorly on files containing heterogeneous data.