Modulo operation

Encyclopedia

In computing

, the

of division

of one number by another.

Given two positive numbers, (the dividend

) and (the divisor

),

for an older and related convention applied in number theory

.

There are various ways of defining a remainder, and computers and calculators have various ways of storing and representing numbers, so what exactly constitutes the result of a modulo operation depends on the programming language

and/or the underlying hardware

.

In nearly all computing systems, the quotient

and the remainder satisfy

This means, that if the remainder is nonzero, there are two possible choices for the remainder, one negative and the other positive, and there are also two possible choices for the quotient. Usually, in number theory, the positive remainder is always chosen, but programming languages choose depending on the language and the signs of

Many implementations use

Knuth

described

Here the quotient is always rounded downwards (even if it is already negative) and the remainder has the same sign as the divisor.

Raymond T. Boute introduces the

. This definition is marked as "Always positive" in the table. Let

Two corollaries are that

or, equivalently,

As described by Leijen,

Common Lisp also defines round- and ceiling-division where the quotient is given by , .

IEEE 754 defines a remainder function where the quotient is rounded according to the round to nearest convention.

For example, to test whether an integer is odd, one might be inclined to test whether the remainder by 2 is equal to 1:

Computing

Computing is usually defined as the activity of using and improving computer hardware and software. It is the computer-specific part of information technology...

, the

**modulo**operation finds the remainderRemainder

In arithmetic, the remainder is the amount "left over" after the division of two integers which cannot be expressed with an integer quotient....

of division

Division (mathematics)

right|thumb|200px|20 \div 4=5In mathematics, especially in elementary arithmetic, division is an arithmetic operation.Specifically, if c times b equals a, written:c \times b = a\,...

of one number by another.

Given two positive numbers, (the dividend

Dividend

Dividends are payments made by a corporation to its shareholder members. It is the portion of corporate profits paid out to stockholders. When a corporation earns a profit or surplus, that money can be put to two uses: it can either be re-invested in the business , or it can be distributed to...

) and (the divisor

Divisor

In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...

),

*a***modulo***n*(abbreviated as*a*mod*n*) can be thought of as the remainder, on division of*a*by*n*. For instance, the expression "5 mod 4" would evaluate to 1 because 5 divided by 4 leaves a remainder of 1, while "9 mod 3" would evaluate to 0 because the division of 9 by 3 leaves a remainder of 0; there is nothing to subtract from 9 after multiplying 3 times 3. (Notice that doing the division with a calculator won't show you the result referred to here by this operation, the quotient will be expressed as a decimal.) When either or is negative, this naive definition breaks down and programming languages differ in how these values are defined. Although typically performed with*a*and*n*both being integers, many computing systems allow other types of numeric operands. The range of numbers for an integer modulo of*n*is 0 to*n*- 1. (*n*mod 1 is always 0;*n*mod 0 is undefined, possibly resulting in a "Division by zero" error in computer programming languages) See modular arithmeticModular arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....

for an older and related convention applied in number theory

Number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

.

## Remainder calculation for the modulo operation

Language | Operator | Result has the same sign as |
---|---|---|

ActionScript ActionScript ActionScript is an object-oriented language originally developed by Macromedia Inc. . It is a dialect of ECMAScript , and is used primarily for the development of websites and software targeting the Adobe Flash Player platform, used on Web pages in the form of... |
% |
Dividend |

Ada Ada (programming language) Ada is a structured, statically typed, imperative, wide-spectrum, and object-oriented high-level computer programming language, extended from Pascal and other languages... |
mod |
Divisor |

rem |
Dividend | |

ASP | Mod |
Not defined |

ALGOL-68 | %× |
Always positive |

AMPL AMPL AMPL, an acronym for "A Mathematical Programming Language", is an algebraic modeling language for describing and solving high-complexity problems for large-scale mathematical computation AMPL, an acronym for "A Mathematical Programming Language", is an algebraic modeling language for describing and... |
mod |
Dividend |

AppleScript AppleScript AppleScript is a scripting language created by Apple Inc. and built into Macintosh operating systems since System 7. The term "AppleScript" may refer to the scripting system itself, or to particular scripts that are written in the AppleScript language.... |
mod |
Dividend |

BASIC BASIC BASIC is a family of general-purpose, high-level programming languages whose design philosophy emphasizes ease of use - the name is an acronym from Beginner's All-purpose Symbolic Instruction Code.... |
Mod |
Not defined |

bc Bc programming language bc, for bench calculator, is "an arbitrary precision calculator language" with syntax similar to the C programming language. bc is typically used as either a mathematical scripting language or as an interactive mathematical shell.... |
% |
Dividend |

bash | % |
Dividend |

C C (programming language) C is a general-purpose computer programming language developed between 1969 and 1973 by Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix operating system.... (ISO 1990) |
% |
Implementation defined |

C (ISO 1999) C99 C99 is a modern dialect of the C programming language. It extends the previous version with new linguistic and library features, and helps implementations make better use of available computer hardware and compiler technology.-History:... |
% |
Dividend |

C++ C++ C++ is a statically typed, free-form, multi-paradigm, compiled, general-purpose programming language. It is regarded as an intermediate-level language, as it comprises a combination of both high-level and low-level language features. It was developed by Bjarne Stroustrup starting in 1979 at Bell... |
% |
Implementation defined |

C# | % |
Dividend |

CLARION | % |
Dividend |

Clojure Clojure Clojure |closure]]") is a recent dialect of the Lisp programming language created by Rich Hickey. It is a general-purpose language supporting interactive development that encourages a functional programming style, and simplifies multithreaded programming.... |
mod |
Divisor |

ColdFusion ColdFusion In computing, ColdFusion is the name of a commercial rapid application development platform invented by Jeremy and JJ Allaire in 1995. ColdFusion was originally designed to make it easier to connect simple HTML pages to a database, by version 2 it had... |
% |
Dividend |

Common Lisp Common Lisp Common Lisp, commonly abbreviated CL, is a dialect of the Lisp programming language, published in ANSI standard document ANSI INCITS 226-1994 , . From the ANSI Common Lisp standard the Common Lisp HyperSpec has been derived for use with web browsers... |
mod |
Divisor |

rem |
Dividend | |

D D (programming language) The D programming language is an object-oriented, imperative, multi-paradigm, system programming language created by Walter Bright of Digital Mars. It originated as a re-engineering of C++, but even though it is mainly influenced by that language, it is not a variant of C++... |
% |
Dividend |

Eiffel Eiffel (programming language) Eiffel is an ISO-standardized, object-oriented programming language designed by Bertrand Meyer and Eiffel Software. The design of the language is closely connected with the Eiffel programming method... |
\\ |
Dividend |

Erlang | rem |
Dividend |

Euphoria | mod |
Divisor |

remainder |
Dividend | |

FileMaker FileMaker FileMaker Pro is a cross-platform relational database application from FileMaker Inc., formerly Claris, a subsidiary of Apple Inc. It integrates a database engine with a GUI-based interface, allowing users to modify the database by dragging new elements into layouts, screens, or forms... |
Mod |
Divisor |

Fortran Fortran Fortran is a general-purpose, procedural, imperative programming language that is especially suited to numeric computation and scientific computing... |
mod |
Dividend |

modulo |
Divisor | |

GML (Game Maker) Game Maker Language Game Maker Language is a scripting language developed for use with a computer game creation application called Game Maker. It was originally created by Mark Overmars to supplement the drag-and-drop action system used in Game Maker... |
mod |
Dividend |

Go Go (programming language) Go is a compiled, garbage-collected, concurrent programming language developed by Google Inc.The initial design of Go was started in September 2007 by Robert Griesemer, Rob Pike, and Ken Thompson. Go was officially announced in November 2009. In May 2010, Rob Pike publicly stated that Go was being... |
% |
Dividend |

Haskell Haskell (programming language) Haskell is a standardized, general-purpose purely functional programming language, with non-strict semantics and strong static typing. It is named after logician Haskell Curry. In Haskell, "a function is a first-class citizen" of the programming language. As a functional programming language, the... |
mod |
Divisor |

rem |
Dividend | |

J J (programming language) The J programming language, developed in the early 1990s by Kenneth E. Iverson and Roger Hui, is a synthesis of APL and the FP and FL function-level languages created by John Backus.... |
Divisor | |

Java Java (programming language) Java is a programming language originally developed by James Gosling at Sun Microsystems and released in 1995 as a core component of Sun Microsystems' Java platform. The language derives much of its syntax from C and C++ but has a simpler object model and fewer low-level facilities... |
% |
Dividend |

JavaScript JavaScript JavaScript is a prototype-based scripting language that is dynamic, weakly typed and has first-class functions. It is a multi-paradigm language, supporting object-oriented, imperative, and functional programming styles.... |
% |
Dividend |

Just Basic Just BASIC Just BASIC is a dialect of the highly popular programming language BASIC of the 1970s, for 32-bit computer systems using Windows. It's the freeware version of Liberty BASIC, popular since 1992. Just BASIC development began in 2001; first public release was in 2004... |
MOD |
Dividend |

Lua 5 | % |
Divisor |

Lua 4 | mod(x,y) |
Divisor |

Liberty Basic Liberty BASIC Liberty BASIC is a commercial computer programming language and integrated development environment . It has an interpreter developed in Smalltalk, which recognizes its own dialect of the BASIC programming language... |
MOD |
Dividend |

MathCad MathCad Mathcad is computer software primarily intended for the verification, validation, documentation and re-use of engineering calculations. First introduced in 1986 on DOS, it was the first to introduce live editing of typeset mathematical notation, combined with its automatic computations... |
mod(x,y) |
Divisor |

Maple (software) Maple (software) Maple is a general-purpose commercial computer algebra system. It was first developed in 1980 by the Symbolic Computation Group at the University of Waterloo in Waterloo, Ontario, Canada.... |
e mod m |
Divisor |

Mathematica Mathematica Mathematica is a computational software program used in scientific, engineering, and mathematical fields and other areas of technical computing... |
Mod |
Divisor |

Microsoft Excel Microsoft Excel Microsoft Excel is a proprietary commercial spreadsheet application written and distributed by Microsoft for Microsoft Windows and Mac OS X. It features calculation, graphing tools, pivot tables, and a macro programming language called Visual Basic for Applications... |
=MOD |
Divisor |

Minitab Minitab Minitab is a statistics package. It was developed at the Pennsylvania State University by researchers Barbara F. Ryan, Thomas A. Ryan, Jr., and Brian L. Joiner in 1972... |
MOD |
Divisor |

MATLAB MATLAB MATLAB is a numerical computing environment and fourth-generation programming language. Developed by MathWorks, MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages,... |
mod |
Divisor |

rem |
Dividend | |

Oberon Oberon (programming language) Oberon is a programming language created in 1986 by Professor Niklaus Wirth and his associates at ETH Zurich in Switzerland. It was developed as part of the implementation of the Oberon operating system... |
MOD |
Divisor |

Objective Caml Objective Caml OCaml , originally known as Objective Caml, is the main implementation of the Caml programming language, created by Xavier Leroy, Jérôme Vouillon, Damien Doligez, Didier Rémy and others in 1996... |
mod |
Dividend |

Occam Occam (programming language) occam is a concurrent programming language that builds on the Communicating Sequential Processes process algebra, and shares many of its features. It is named after William of Ockham of Occam's Razor fame.... |
\ |
Dividend |

Pascal (Delphi) Pascal (programming language) Pascal is an influential imperative and procedural programming language, designed in 1968/9 and published in 1970 by Niklaus Wirth as a small and efficient language intended to encourage good programming practices using structured programming and data structuring.A derivative known as Object Pascal... |
mod |
Dividend |

Pascal (ISO-7185 and ISO-10206) Pascal (programming language) Pascal is an influential imperative and procedural programming language, designed in 1968/9 and published in 1970 by Niklaus Wirth as a small and efficient language intended to encourage good programming practices using structured programming and data structuring.A derivative known as Object Pascal... |
mod |
Always positive |

Perl Perl Perl is a high-level, general-purpose, interpreted, dynamic programming language. Perl was originally developed by Larry Wall in 1987 as a general-purpose Unix scripting language to make report processing easier. Since then, it has undergone many changes and revisions and become widely popular... |
% |
Divisor |

PHP PHP PHP is a general-purpose server-side scripting language originally designed for web development to produce dynamic web pages. For this purpose, PHP code is embedded into the HTML source document and interpreted by a web server with a PHP processor module, which generates the web page document... |
% |
Dividend |

PIC Basic Pro | \\ |
Dividend |

PL/I PL/I PL/I is a procedural, imperative computer programming language designed for scientific, engineering, business and systems programming applications... |
mod |
Divisor (ANSI PL/I) |

PowerBuilder PowerBuilder PowerBuilder is an integrated development environment owned by Sybase, a division of SAP. It has been in use since 1991, peaking around 1998 with around 100,000 users.... |
mod(x,y) |
? |

PowerShell | % |
Dividend |

Progress Progress -History:*Progress , the idea that the world can become increasingly better in terms of science, technology, modernization, liberty, democracy, quality of life, etc.... |
modulo |
Dividend |

Prolog Prolog Prolog is a general purpose logic programming language associated with artificial intelligence and computational linguistics.Prolog has its roots in first-order logic, a formal logic, and unlike many other programming languages, Prolog is declarative: the program logic is expressed in terms of... (ISO 1995) |
mod |
Divisor |

rem |
Dividend | |

Python Python (programming language) Python is a general-purpose, high-level programming language whose design philosophy emphasizes code readability. Python claims to "[combine] remarkable power with very clear syntax", and its standard library is large and comprehensive... |
% |
Divisor |

RealBasic REALbasic Realbasic is the object-oriented dialect of the BASIC programming language used in Real Studio, a programming environment, developed and commercially marketed by Real Software, Inc of Austin, Texas for Mac OS X, Microsoft Windows, 32-bit x86 Linux and the web.- Language features :RB is a strongly... |
MOD |
Dividend |

R R (programming language) R is a programming language and software environment for statistical computing and graphics. The R language is widely used among statisticians for developing statistical software, and R is widely used for statistical software development and data analysis.... |
%% |
Divisor |

RPG | %REM |
Dividend |

Ruby Ruby (programming language) Ruby is a dynamic, reflective, general-purpose object-oriented programming language that combines syntax inspired by Perl with Smalltalk-like features. Ruby originated in Japan during the mid-1990s and was first developed and designed by Yukihiro "Matz" Matsumoto... |
%, modulus |
Divisor |

remainder |
Dividend | |

Scheme | modulo |
Divisor |

remainder |
Dividend | |

Scheme R^{6}RS |
mod |
Always positive |

mod0 |
Closest to zero | |

SenseTalk SenseTalk SenseTalk is an English-like scripting language derived from the HyperTalk language used in HyperCard. SenseTalk was originally developed as the scripting language within the HyperSense multimedia authoring application on the NeXTStep and OpenStep platforms... |
modulo |
Divisor |

rem |
Dividend | |

Smalltalk Smalltalk Smalltalk is an object-oriented, dynamically typed, reflective programming language. Smalltalk was created as the language to underpin the "new world" of computing exemplified by "human–computer symbiosis." It was designed and created in part for educational use, more so for constructionist... |
\\ |
Divisor |

SQL SQL SQL is a programming language designed for managing data in relational database management systems .... (SQL:1999 SQL:1999 SQL:1999 was the fourth revision of the SQL database query language. The latest revision of the standard is SQL:2008.-Summary:The SQL:1999 standard, also known as SQL3, was published in 1999. Unlike previous editions, the standard's name used a colon instead of a hyphen for consistency with the... ) |
mod(x,y) |
Dividend |

Standard ML Standard ML Standard ML is a general-purpose, modular, functional programming language with compile-time type checking and type inference. It is popular among compiler writers and programming language researchers, as well as in the development of theorem provers.SML is a modern descendant of the ML... |
mod |
Divisor |

Int.rem |
Dividend | |

Stata Stata Stata is a general-purpose statistical software package created in 1985 by StataCorp. It is used by many businesses and academic institutions around the world... |
mod(x,y) |
Always positive |

Tcl Tcl Tcl is a scripting language created by John Ousterhout. Originally "born out of frustration", according to the author, with programmers devising their own languages intended to be embedded into applications, Tcl gained acceptance on its own... |
% |
Divisor |

Torque Game Engine Torque Game Engine The Torque Game Engine, or TGE, is a 3D computer game engine originally developed by Dynamix for the 2001 FPS Tribes 2. The Torque engine and its many derivative products are available for license from GarageGames, a company formed by many members of the Tribes 2 team at Dynamix... |
% |
Dividend |

Turing | mod |
Divisor |

Verilog Verilog In the semiconductor and electronic design industry, Verilog is a hardware description language used to model electronic systems. Verilog HDL, not to be confused with VHDL , is most commonly used in the design, verification, and implementation of digital logic chips at the register-transfer level... (2001) |
% |
Dividend |

VHDL | mod |
Divisor |

rem |
Dividend | |

Visual Basic Visual Basic Visual Basic is the third-generation event-driven programming language and integrated development environment from Microsoft for its COM programming model... |
Mod |
Dividend |

x86 Assembly X86 assembly language x86 assembly language is a family of backward-compatible assembly languages, which provide some level of compatibility all the way back to the Intel 8008. x86 assembly languages are used to produce object code for the x86 class of processors, which includes Intel's Core series and AMD's Phenom and... |
IDIV |
Dividend |

Language | Operator | Result has the same sign as |
---|---|---|

C C (programming language) C is a general-purpose computer programming language developed between 1969 and 1973 by Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix operating system.... (ISO 1990) |
fmod |
? |

C (ISO 1999) C99 C99 is a modern dialect of the C programming language. It extends the previous version with new linguistic and library features, and helps implementations make better use of available computer hardware and compiler technology.-History:... |
fmod |
Dividend |

remainder |
Closest to zero | |

C++ C++ C++ is a statically typed, free-form, multi-paradigm, compiled, general-purpose programming language. It is regarded as an intermediate-level language, as it comprises a combination of both high-level and low-level language features. It was developed by Bjarne Stroustrup starting in 1979 at Bell... (ISO 1998) |
std::fmod |
? |

C++ (ISO 2011) | std::fmod |
Dividend |

std::remainder |
Closest to zero | |

C# | % |
Dividend |

Common Lisp Common Lisp Common Lisp, commonly abbreviated CL, is a dialect of the Lisp programming language, published in ANSI standard document ANSI INCITS 226-1994 , . From the ANSI Common Lisp standard the Common Lisp HyperSpec has been derived for use with web browsers... |
mod |
Divisor |

rem |
Dividend | |

D D (programming language) The D programming language is an object-oriented, imperative, multi-paradigm, system programming language created by Walter Bright of Digital Mars. It originated as a re-engineering of C++, but even though it is mainly influenced by that language, it is not a variant of C++... |
% |
? |

Fortran Fortran Fortran is a general-purpose, procedural, imperative programming language that is especially suited to numeric computation and scientific computing... |
mod |
Dividend |

modulo |
Divisor | |

Go Go (programming language) Go is a compiled, garbage-collected, concurrent programming language developed by Google Inc.The initial design of Go was started in September 2007 by Robert Griesemer, Rob Pike, and Ken Thompson. Go was officially announced in November 2009. In May 2010, Rob Pike publicly stated that Go was being... |
math.Fmod |
Dividend |

Haskell Haskell (programming language) Haskell is a standardized, general-purpose purely functional programming language, with non-strict semantics and strong static typing. It is named after logician Haskell Curry. In Haskell, "a function is a first-class citizen" of the programming language. As a functional programming language, the... (GHC) |
Data.Fixed.mod' |
Divisor |

Java Java (programming language) Java is a programming language originally developed by James Gosling at Sun Microsystems and released in 1995 as a core component of Sun Microsystems' Java platform. The language derives much of its syntax from C and C++ but has a simpler object model and fewer low-level facilities... |
% |
Dividend |

JavaScript JavaScript JavaScript is a prototype-based scripting language that is dynamic, weakly typed and has first-class functions. It is a multi-paradigm language, supporting object-oriented, imperative, and functional programming styles.... |
% |
Dividend |

Objective Caml Objective Caml OCaml , originally known as Objective Caml, is the main implementation of the Caml programming language, created by Xavier Leroy, Jérôme Vouillon, Damien Doligez, Didier Rémy and others in 1996... |
mod_float |
Dividend |

Perl Perl Perl is a high-level, general-purpose, interpreted, dynamic programming language. Perl was originally developed by Larry Wall in 1987 as a general-purpose Unix scripting language to make report processing easier. Since then, it has undergone many changes and revisions and become widely popular... |
POSIX::fmod |
Dividend |

PHP PHP PHP is a general-purpose server-side scripting language originally designed for web development to produce dynamic web pages. For this purpose, PHP code is embedded into the HTML source document and interpreted by a web server with a PHP processor module, which generates the web page document... |
fmod |
Dividend |

Python Python (programming language) Python is a general-purpose, high-level programming language whose design philosophy emphasizes code readability. Python claims to "[combine] remarkable power with very clear syntax", and its standard library is large and comprehensive... |
% |
Divisor |

math.fmod |
Dividend | |

Ruby Ruby (programming language) Ruby is a dynamic, reflective, general-purpose object-oriented programming language that combines syntax inspired by Perl with Smalltalk-like features. Ruby originated in Japan during the mid-1990s and was first developed and designed by Yukihiro "Matz" Matsumoto... |
%, modulus |
Divisor |

remainder |
Dividend | |

Scheme R^{6}RS |
flmod |
Always positive |

flmod0 |
Closest to zero | |

Standard ML Standard ML Standard ML is a general-purpose, modular, functional programming language with compile-time type checking and type inference. It is popular among compiler writers and programming language researchers, as well as in the development of theorem provers.SML is a modern descendant of the ML... |
Real.rem |
Dividend |

There are various ways of defining a remainder, and computers and calculators have various ways of storing and representing numbers, so what exactly constitutes the result of a modulo operation depends on the programming language

Programming language

A programming language is an artificial language designed to communicate instructions to a machine, particularly a computer. Programming languages can be used to create programs that control the behavior of a machine and/or to express algorithms precisely....

and/or the underlying hardware

Computer hardware

Personal computer hardware are component devices which are typically installed into or peripheral to a computer case to create a personal computer upon which system software is installed including a firmware interface such as a BIOS and an operating system which supports application software that...

.

In nearly all computing systems, the quotient

Quotient

In mathematics, a quotient is the result of division. For example, when dividing 6 by 3, the quotient is 2, while 6 is called the dividend, and 3 the divisor. The quotient further is expressed as the number of times the divisor divides into the dividend e.g. The quotient of 6 and 2 is also 3.A...

and the remainder satisfy

This means, that if the remainder is nonzero, there are two possible choices for the remainder, one negative and the other positive, and there are also two possible choices for the quotient. Usually, in number theory, the positive remainder is always chosen, but programming languages choose depending on the language and the signs of

*a*and*n*. However, Pascal and Algol68 give a positive remainder (or 0) even for negative divisors, and some programming languages, such as C89, don't even define a result if either of*n*or*a*is negative. See the table for details.*a*modulo 0 is undefined in the majority of systems, although some do define it to be*a*.Many implementations use

**truncated division**where the quotient is defined by truncationTruncation

In mathematics and computer science, truncation is the term for limiting the number of digits right of the decimal point, by discarding the least significant ones.For example, consider the real numbersThe result would be:- Truncation and floor function :...

*q*= trunc(*a*/*n*), in other words it is the first integer in the direction of 0 from the exact rational quotient, and the remainder by*r*=*a*−*n**q*. Informally speaking the quotient is "rounded towards zero", and the remainder therefore has the same sign as the dividend.Knuth

Donald Knuth

Donald Ervin Knuth is a computer scientist and Professor Emeritus at Stanford University.He is the author of the seminal multi-volume work The Art of Computer Programming. Knuth has been called the "father" of the analysis of algorithms...

described

**floored division**where the quotient is defined by the floor functionFloor function

In mathematics and computer science, the floor and ceiling functions map a real number to the largest previous or the smallest following integer, respectively...

*q*=floor(*a*/*n*) and the remainder*r*isHere the quotient is always rounded downwards (even if it is already negative) and the remainder has the same sign as the divisor.

Raymond T. Boute introduces the

**Euclidean definition**, which is the one in which the remainder is always positive or 0, and is therefore consistent with the division algorithmDivision algorithm

In mathematics, and more particularly in arithmetic, the usual process of division of integers producing a quotient and a remainder can be specified precisely by a theorem stating that these exist uniquely with given properties. An integer division algorithm is any effective method for producing...

. This definition is marked as "Always positive" in the table. Let

*q*be the integer quotient of*a*and*n*, then:Two corollaries are that

or, equivalently,

As described by Leijen,

- Boute argues that Euclidean division is superior to the other ones in terms of regularity and useful mathematical properties, although floored division, promoted by Knuth, is also a good definition. Despite its widespread use, truncated division is shown to be inferior to the other definitions.

Common Lisp also defines round- and ceiling-division where the quotient is given by , .

IEEE 754 defines a remainder function where the quotient is rounded according to the round to nearest convention.

## Common pitfalls

When the result of a modulo operation has the sign of the dividend, it can sometimes lead to surprising mistakes:For example, to test whether an integer is odd, one might be inclined to test whether the remainder by 2 is equal to 1: