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.

A stellarator configuration utilizing an optimized modular coil set designed to simultaneously achieve high plasma, low Pfirsch-Schluter currents and good confinement of energetic particles ; i. e., alpha particles for reactor scenarios.

During the Brain Salad Surgery tour of 1974 ( one show of which was documented on the 3-LP set, Welcome Back My Friends, to the Show That Never Ends ), Emerson's keyboard setup included the Hammond C-3 organ, run through multiple Leslie speakers driven by HiWatt guitar amplifiers, the Moog 3C modular synthesiser ( modified by addition of various modules and an oscilloscope ) with ribbon controller, a Steinway concert grand piano with a Moog Minimoog synthesiser on top of it ( used for the steel drum part on Karn Evil 9, 2nd Impression ), an upright acoustic-electric piano that was used for honky-tonk piano sounds, a Hohner Clavinet and another Moog Minimoog synthesiser.

They include a programmable ' Brick ' computer that controls the system, a set of modular sensors and motors, and LEGO parts from the Technics line to create the mechanical systems.

Over 35 years, Tandem's main NonStop product line has grown and evolved in an upward-compatible way from the initial T / 16 fault-tolerant system, with three major changes to date to its top-level modular architecture or its programming-level instruction set architecture.

It is made with modular raised square textured granite slabs ( each slab costing approximately $ 1, 500: $ 1, 000 materials plus $ 500 labour ), features a diagonally running zinc canopy along the northern hypotenuse of the " square ", a movable plinth which serves as a stage for concerts and other performances, a row of lighted fountains set directly into the pavement, a row of small trees along the southern edge, a transparent canopy over the plinth, and a new entrance to Dundas subway station below.

DSDL is a multipart standard defining a modular set of specifications for describing the document structures, data types, and data relationships in structured information resources.

As a result he changed his major subject in the subsequent two years to acting, constructing his modular Independent Studies Degree around acting, set design studying Stanislavski and Brecht, and minor modules in art and visual graphics.

Supreme Commander set out to be the ' most customisable game ever ' and as such included a mod manager which allowed for modular modding, having several mods on at once.

The Hicom / HiPath series are a modular set of phone systems which come in several shapes and sizes, from the smallest wall-mountable units which support a few dozen clients to the largest modular block units which take multiple cards for digital ( ISDN ), POTS ( analog ), and VoIP cards.

Let M < sub > m </ sub > be the set of 2 × 2 integral matrices with determinant m and Γ = M < sub > 1 </ sub > be the full modular group SL ( 2, Z ).

* When a modular unit is finished, a whole set of tests are performed and if quality standards are met, a unique number and EC stamp is attached to and on the unit.

The SLS used a set of common modular components with solid fuel boosters and hydrogen / oxygen upper stages to allow a wide variety of launch configurations and payload weights.

Additionally, the reduction in radio size, cost, and power has enabled the mesh nodes to become more modular — one node or device now can contain multiple radio cards or modules, allowing the nodes to be customized to handle a unique set of functions and frequency bands.

This would set the division to become the Army ’ s first fully digitized modular division.

* Malitte ( modular furniture set designed between 1966 and 1968 ) and La Caza de Adolescentes ( 1968 )

Some 4 ton trucks for example had a modular set of fuel tanks with dispensing nozzles to increase the speed at which tanks could be refueled during combat ( much faster and cleaner than emptying Jerrycans of fuel into the 220 gallon fuel tanks ).

With money earned from performing as an underwater escape artiste in the Berlin Circus he set up Shotline Steel Swimming Pools, a company which supplies modular swimming pools to schools in the UK.

In March 1998 Adtranz presented a set of new product brands for modular product platforms, with designs that can be adapted for the specific requirements of different customers: the Innovia guided transport vehicle, the Incentro low floor tram, the Itino diesel multiple unit, the Crusaris medium-high speed train ( based on the GMB Class 71 flytoget trains ), and the Octeon electric locomotive.

Structured cabling design and installation is governed by a set of standards that specify wiring data centers, offices, and apartment buildings for data or voice communications using various kinds of cable, most commonly category 5e ( CAT-5e ), category 6 ( CAT-6 ), and fibre optic cabling and modular connectors.

; Clerezza: An OSGi-based modular application and set of components ( bundles ) for building RESTFul Semantic Web applications and services.

; Stanbol: A modular software stack and reusable set of components for semantic content management

He wrote almost a hundred other papers, mostly on finite group theory, character theory ( in particular introducing the concept of a coherent set of characters ), and modular representation theory.