Modulo

Template:About Template:Short description In computing and mathematics, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, the latter being called the modulus of the operation.

Given two positive numbers Template:Math and Template:Math, Template:Math modulo Template:Math (often abbreviated as Template:Math) is the remainder of the Euclidean division of Template:Math by Template:Math, where Template:Math is the dividend and Template:Math is the divisor.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

For example, the expression "5 mod 2" evaluates to 1, because 5 divided by 2 has a quotient of 2 and a remainder of 1, while "9 mod 3" would evaluate to 0, because 9 divided by 3 has a quotient of 3 and a remainder of 0.

Although typically performed with Template:Math and Template:Math both being integers, many computing systems now allow other types of numeric operands. The range of values for an integer modulo operation of Template:Math is 0 to Template:Math. Template:Math mod 1 is always 0.

When exactly one of Template:Math or Template:Math is negative, the basic definition breaks down, and programming languages differ in how these values are defined.

Variants of the definitionEdit

In mathematics, the result of the modulo operation is an equivalence class, and any member of the class may be chosen as representative; however, the usual representative is the least positive residue, the smallest non-negative integer that belongs to that class (i.e., the remainder of the Euclidean division).<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> However, other conventions are possible. Computers and calculators have various ways of storing and representing numbers; thus their definition of the modulo operation depends on the programming language or the underlying hardware.

In nearly all computing systems, the quotient Template:Math and the remainder Template:Math of Template:Math divided by Template:Math satisfy the following conditions: Template:NumBlk

This still leaves a sign ambiguity if the remainder is non-zero: two possible choices for the remainder occur, one negative and the other positive; that choice determines which of the two consecutive quotients must be used to satisfy equation (1). In number theory, the positive remainder is always chosen, but in computing, programming languages choose depending on the language and the signs of Template:Math or Template:Math.Template:Efn Standard Pascal and ALGOL 68, for example, give a positive remainder (or 0) even for negative divisors, and some programming languages, such as C90, leave it to the implementation when either of Template:Math or Template:Math is negative (see the table under Template:Section link for details). Some systems leave Template:Math modulo 0 undefined, though others define it as Template:Math.

Template:Bulleted list

If both the dividend and divisor are positive, then the truncated, floored, and Euclidean definitions agree. If the dividend is positive and the divisor is negative, then the truncated and Euclidean definitions agree. If the dividend is negative and the divisor is positive, then the floored and Euclidean definitions agree. If both the dividend and divisor are negative, then the truncated and floored definitions agree.

As described by Leijen, Template:Quote

However, truncated division satisfies the identity <math>({-a})/b = {-(a/b)} = a/({-b})</math>.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

NotationEdit

Template:About

Some calculators have a Template:Math function button, and many programming languages have a similar function, expressed as Template:Math, for example. Some also support expressions that use "%", "mod", or "Mod" as a modulo or remainder operator, such as <syntaxhighlight lang="text" class="" style="" inline="1">a % n</syntaxhighlight> or <syntaxhighlight lang="text" class="" style="" inline="1">a mod n</syntaxhighlight>.

For environments lacking a similar function, any of the three definitions above can be used.

Common pitfallsEdit

When the result of a modulo operation has the sign of the dividend (truncated definition), it can lead to surprising mistakes.

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

<syntaxhighlight lang="cpp"> bool is_odd(int n) {

   return n % 2 == 1;

} </syntaxhighlight>

But in a language where modulo has the sign of the dividend, that is incorrect, because when Template:Math (the dividend) is negative and odd, Template:Math mod 2 returns −1, and the function returns false.

One correct alternative is to test that the remainder is not 0 (because remainder 0 is the same regardless of the signs):

<syntaxhighlight lang="cpp"> bool is_odd(int n) {

   return n % 2 != 0;

} </syntaxhighlight>

Or with the binary arithmetic: <syntaxhighlight lang="cpp"> bool is_odd(int n) {

   return n & 1;

} </syntaxhighlight>

Performance issuesEdit

Modulo operations might be implemented such that a division with a remainder is calculated each time. For special cases, on some hardware, faster alternatives exist. For example, the modulo of powers of 2 can alternatively be expressed as a bitwise AND operation (assuming Template:Math is a positive integer, or using a non-truncating definition):

x % 2ⁿ == x & (2ⁿ - 1)

Examples:

In devices and software that implement bitwise operations more efficiently than modulo, these alternative forms can result in faster calculations.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Compiler optimizations may recognize expressions of the form <syntaxhighlight lang="text" class="" style="" inline="1">expression % constant</syntaxhighlight> where <syntaxhighlight lang="text" class="" style="" inline="1">constant</syntaxhighlight> is a power of two and automatically implement them as <syntaxhighlight lang="text" class="" style="" inline="1">expression & (constant-1)</syntaxhighlight>, allowing the programmer to write clearer code without compromising performance. This simple optimization is not possible for languages in which the result of the modulo operation has the sign of the dividend (including C), unless the dividend is of an unsigned integer type. This is because, if the dividend is negative, the modulo will be negative, whereas <syntaxhighlight lang="text" class="" style="" inline="1">expression & (constant-1)</syntaxhighlight> will always be positive. For these languages, the equivalence x % 2ⁿ == x < 0 ? x | ~(2ⁿ - 1) : x & (2ⁿ - 1) has to be used instead, expressed using bitwise OR, NOT and AND operations.

Optimizations for general constant-modulus operations also exist by calculating the division first using the constant-divisor optimization.

Properties (identities)Edit

Template:See also Some modulo operations can be factored or expanded similarly to other mathematical operations. This may be useful in cryptography proofs, such as the Diffie–Hellman key exchange. The properties involving multiplication, division, and exponentiation generally require that Template:Math and Template:Math are integers.

Identity:
- Template:Math.
- Template:Math for all positive integer values of Template:Math.
- If Template:Math is a prime number which is not a divisor of Template:Math, then Template:Math, due to Fermat's little theorem.
Inverse:
- Template:Math.
- Template:Math denotes the modular multiplicative inverse, which is defined if and only if Template:Math and Template:Math are relatively prime, which is the case when the left hand side is defined: Template:Math.
Distributive:
- Template:Math.
- Template:Math.
Division (definition): Template:Math, when the right hand side is defined (that is when Template:Math and Template:Math are coprime), and undefined otherwise.
Inverse multiplication: Template:Math.

In programming languagesEdit

Template:Sticky header

Modulo operators in various programming languages
Language	Operator	Integer	Floating-point	Definition
ABAP	<syntaxhighlight lang="text" class="" style="" inline="1">MOD</syntaxhighlight>	Template:Yes	Template:Yes	Euclidean
ActionScript	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>	Template:Yes	Template:No	Truncated
Ada	<syntaxhighlight lang="text" class="" style="" inline="1">mod</syntaxhighlight>	Template:Yes	Template:No	Floored<ref name="Ada">Template:Cite book</ref>
Ada	<syntaxhighlight lang="text" class="" style="" inline="1">rem</syntaxhighlight>	Template:Yes	Template:No	Truncated<ref name="Ada" />
ALGOL 68	<syntaxhighlight lang="text" class="" style="" inline="1">÷×</syntaxhighlight>, <syntaxhighlight lang="text" class="" style="" inline="1">mod</syntaxhighlight>	Template:Yes	Template:No	Euclidean
AMPL	<syntaxhighlight lang="text" class="" style="" inline="1">mod</syntaxhighlight>	Template:Yes	Template:No	Truncated
APL	`\|`Template:Efn	Template:Yes	Template:Yes	Floored
AppleScript	<syntaxhighlight lang="text" class="" style="" inline="1">mod</syntaxhighlight>	Template:Yes	Template:No	Truncated
AutoLISP	<syntaxhighlight lang="text" class="" style="" inline="1">(rem d n)</syntaxhighlight>	Template:Yes	Template:No	Truncated
AWK	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>	Template:Yes	Template:No	Truncated
bash	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>	Template:Yes	Template:No	Truncated
BASIC	<syntaxhighlight lang="text" class="" style="" inline="1">Mod</syntaxhighlight>	Template:Yes	Template:No	Varies by implementation
bc	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>	Template:Yes	Template:No	Truncated
C Template:Break C++	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>, <syntaxhighlight lang="text" class="" style="" inline="1">div</syntaxhighlight>	Template:Yes	Template:No	TruncatedTemplate:Efn
	<syntaxhighlight lang="text" class="" style="" inline="1">fmod</syntaxhighlight> (C)Template:Break<syntaxhighlight lang="text" class="" style="" inline="1">std::fmod</syntaxhighlight> (C++)	Template:No	Template:Yes	Truncated<ref>Template:Cite book</ref>
	<syntaxhighlight lang="text" class="" style="" inline="1">remainder</syntaxhighlight> (C)Template:Break<syntaxhighlight lang="text" class="" style="" inline="1">std::remainder</syntaxhighlight> (C++)	Template:No	Template:Yes	Rounded
C#	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>	Template:Yes	Template:Yes	Truncated
C#	<syntaxhighlight lang="text" class="" style="" inline="1">Math.IEEERemainder</syntaxhighlight>	Template:No	Template:Yes	citation	CitationClass=web }}</ref>
Clarion	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>	Template:Yes	Template:No	Truncated
Clean	<syntaxhighlight lang="text" class="" style="" inline="1">rem</syntaxhighlight>	Template:Yes	Template:No	Truncated
Clojure	<syntaxhighlight lang="text" class="" style="" inline="1">mod</syntaxhighlight>	Template:Yes	Template:No	citation	CitationClass=web }}</ref>
Clojure	<syntaxhighlight lang="text" class="" style="" inline="1">rem</syntaxhighlight>	Template:Yes	Template:No	citation	CitationClass=web }}</ref>
COBOL	<syntaxhighlight lang="text" class="" style="" inline="1">FUNCTION MOD</syntaxhighlight>	Template:Yes	Template:No	Floored<ref name="isocobol">Template:Cite book</ref>
COBOL	<syntaxhighlight lang="text" class="" style="" inline="1">FUNCTION REM</syntaxhighlight>	Template:Yes	Template:Yes	Truncated<ref name="isocobol"/>
CoffeeScript	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>	Template:Yes	Template:No	Truncated
CoffeeScript	<syntaxhighlight lang="text" class="" style="" inline="1">%%</syntaxhighlight>	Template:Yes	Template:No	Floored<ref name="CoffeeScript">CoffeeScript operators</ref>
ColdFusion	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>, <syntaxhighlight lang="text" class="" style="" inline="1">MOD</syntaxhighlight>	Template:Yes	Template:No	Truncated
Common Intermediate Language	<syntaxhighlight lang="text" class="" style="" inline="1">rem</syntaxhighlight> (signed)	Template:Yes	Template:Yes	Truncated<ref>Template:Cite book</ref>
Common Intermediate Language	<syntaxhighlight lang="text" class="" style="" inline="1">rem.un</syntaxhighlight> (unsigned)	Template:Yes	Template:No	Template:N/A
Common Lisp	<syntaxhighlight lang="text" class="" style="" inline="1">mod</syntaxhighlight>	Template:Yes	Template:Yes	Floored
Common Lisp	<syntaxhighlight lang="text" class="" style="" inline="1">rem</syntaxhighlight>	Template:Yes	Template:Yes	Truncated
Crystal	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>, <syntaxhighlight lang="text" class="" style="" inline="1">modulo</syntaxhighlight>	Template:Yes	Template:Yes	Floored
Crystal	<syntaxhighlight lang="text" class="" style="" inline="1">remainder</syntaxhighlight>	Template:Yes	Template:Yes	Truncated
CSS	<syntaxhighlight lang="text" class="" style="" inline="1">mod()</syntaxhighlight>	Template:Yes	Template:Yes	citation	CitationClass=web }}</ref>
CSS	<syntaxhighlight lang="text" class="" style="" inline="1">rem()</syntaxhighlight>	Template:Yes	Template:Yes	citation	CitationClass=web }}</ref>
D	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>	Template:Yes	Template:Yes	citation	CitationClass=web }}</ref>
Dart	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>	Template:Yes	Template:Yes	citation	CitationClass=web }}</ref>
Dart	<syntaxhighlight lang="text" class="" style="" inline="1">remainder()</syntaxhighlight>	Template:Yes	Template:Yes	citation	CitationClass=web }}</ref>
Eiffel	<syntaxhighlight lang="text" class="" style="" inline="1">\\</syntaxhighlight>	Template:Yes	Template:No	Truncated
Elixir	<syntaxhighlight lang="text" class="" style="" inline="1">rem/2</syntaxhighlight>	Template:Yes	Template:No	citation	CitationClass=web }}</ref>
Elixir	<syntaxhighlight lang="text" class="" style="" inline="1">Integer.mod/2</syntaxhighlight>	Template:Yes	Template:No	citation	CitationClass=web }}</ref>
Elm	<syntaxhighlight lang="text" class="" style="" inline="1">modBy</syntaxhighlight>	Template:Yes	Template:No	citation	CitationClass=web }}</ref>
Elm	<syntaxhighlight lang="text" class="" style="" inline="1">remainderBy</syntaxhighlight>	Template:Yes	Template:No	citation	CitationClass=web }}</ref>
Erlang	<syntaxhighlight lang="text" class="" style="" inline="1">rem</syntaxhighlight>	Template:Yes	Template:No	Truncated
Erlang	<syntaxhighlight lang="text" class="" style="" inline="1">math:fmod/2</syntaxhighlight>	Template:No	Template:Yes	citation	CitationClass=web }}</ref>
Euphoria	<syntaxhighlight lang="text" class="" style="" inline="1">mod</syntaxhighlight>	Template:Yes	Template:No	Floored
Euphoria	<syntaxhighlight lang="text" class="" style="" inline="1">remainder</syntaxhighlight>	Template:Yes	Template:No	Truncated
F#	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>	Template:Yes	Template:Yes	Truncated
F#	<syntaxhighlight lang="text" class="" style="" inline="1">Math.IEEERemainder</syntaxhighlight>	Template:No	Template:Yes	Rounded<ref name=".NET" />
Factor	<syntaxhighlight lang="text" class="" style="" inline="1">mod</syntaxhighlight>	Template:Yes	Template:No	Truncated
FileMaker	<syntaxhighlight lang="text" class="" style="" inline="1">Mod</syntaxhighlight>	Template:Yes	Template:No	Floored
Forth	<syntaxhighlight lang="text" class="" style="" inline="1">mod</syntaxhighlight>	Template:Yes	Template:No	Implementation defined
	<syntaxhighlight lang="text" class="" style="" inline="1">fm/mod</syntaxhighlight>	Template:Yes	Template:No	Floored
	<syntaxhighlight lang="text" class="" style="" inline="1">sm/rem</syntaxhighlight>	Template:Yes	Template:No	Truncated
Fortran	<syntaxhighlight lang="text" class="" style="" inline="1">mod</syntaxhighlight>	Template:Yes	Template:Yes	Truncated
Fortran	<syntaxhighlight lang="text" class="" style="" inline="1">modulo</syntaxhighlight>	Template:Yes	Template:Yes	Floored
Frink	<syntaxhighlight lang="text" class="" style="" inline="1">mod</syntaxhighlight>	Template:Yes	Template:No	Floored
Full BASIC	<syntaxhighlight lang="text" class="" style="" inline="1">MOD</syntaxhighlight>	Template:Yes	Template:Yes	Floored<ref>Template:Cite book</ref>
Full BASIC	<syntaxhighlight lang="text" class="" style="" inline="1">REMAINDER</syntaxhighlight>	Template:Yes	Template:Yes	Truncated<ref>Template:Cite book</ref>
GLSL	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>	Template:Yes	Template:No	citation	CitationClass=web }}</ref>
GLSL	<syntaxhighlight lang="text" class="" style="" inline="1">mod</syntaxhighlight>	Template:No	Template:Yes	citation	CitationClass=web }}</ref>
GameMaker Studio (GML)	<syntaxhighlight lang="text" class="" style="" inline="1">mod</syntaxhighlight>, <syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>	Template:Yes	Template:No	Truncated
GDScript (Godot)	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>	Template:Yes	Template:No	Truncated
	<syntaxhighlight lang="text" class="" style="" inline="1">fmod</syntaxhighlight>	Template:No	Template:Yes	Truncated
	<syntaxhighlight lang="text" class="" style="" inline="1">posmod</syntaxhighlight>	Template:Yes	Template:No	Euclidean
	<syntaxhighlight lang="text" class="" style="" inline="1">fposmod</syntaxhighlight>	Template:No	Template:Yes	Euclidean
Go	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>	Template:Yes	Template:No	citation	CitationClass=web }}</ref>
	<syntaxhighlight lang="text" class="" style="" inline="1">math.Mod</syntaxhighlight>	Template:No	Template:Yes	citation	CitationClass=web }}</ref>
	<syntaxhighlight lang="text" class="" style="" inline="1">big.Int.Mod</syntaxhighlight>	Template:Yes	Template:No	citation	CitationClass=web }}</ref>
	<syntaxhighlight lang="text" class="" style="" inline="1">big.Int.Rem</syntaxhighlight>	Template:Yes	Template:No	citation	CitationClass=web }}</ref>
Groovy	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>	Template:Yes	Template:No	Truncated
Haskell	<syntaxhighlight lang="text" class="" style="" inline="1">mod</syntaxhighlight>	Template:Yes	Template:No	citation	CitationClass=web }}</ref>
	<syntaxhighlight lang="text" class="" style="" inline="1">rem</syntaxhighlight>	Template:Yes	Template:No	Truncated<ref name="Haskell 2010" />
	<syntaxhighlight lang="text" class="" style="" inline="1">Data.Fixed.mod'</syntaxhighlight> (GHC)	Template:No	Template:Yes	Floored
Haxe	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>	Template:Yes	Template:No	Truncated
HLSL	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>	Template:Yes	Template:Yes	citation	CitationClass=web }}</ref>
J	`\|`Template:Efn	Template:Yes	Template:No	Floored
Java	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>	Template:Yes	Template:Yes	Truncated
Java	<syntaxhighlight lang="text" class="" style="" inline="1">Math.floorMod</syntaxhighlight>	Template:Yes	Template:No	Floored
JavaScript Template:Break TypeScript	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>	Template:Yes	Template:Yes	Truncated
Julia	<syntaxhighlight lang="text" class="" style="" inline="1">mod</syntaxhighlight>	Template:Yes	Template:Yes	citation	CitationClass=web }}</ref>
Julia	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>, <syntaxhighlight lang="text" class="" style="" inline="1">rem</syntaxhighlight>	Template:Yes	Template:Yes	citation	CitationClass=web }}</ref>
Kotlin	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>, <syntaxhighlight lang="text" class="" style="" inline="1">rem</syntaxhighlight>	Template:Yes	Template:Yes	citation	CitationClass=web }}</ref>
Kotlin	<syntaxhighlight lang="text" class="" style="" inline="1">mod</syntaxhighlight>	Template:Yes	Template:Yes	citation	CitationClass=web }}</ref>
ksh	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>	Template:Yes	Template:No	Truncated (same as POSIX sh)
ksh	<syntaxhighlight lang="text" class="" style="" inline="1">fmod</syntaxhighlight>	Template:No	Template:Yes	Truncated
LabVIEW	<syntaxhighlight lang="text" class="" style="" inline="1">mod</syntaxhighlight>	Template:Yes	Template:Yes	Truncated
LibreOffice	<syntaxhighlight lang="text" class="" style="" inline="1">=MOD()</syntaxhighlight>	Template:Yes	Template:No	Floored
Logo	<syntaxhighlight lang="text" class="" style="" inline="1">MODULO</syntaxhighlight>	Template:Yes	Template:No	Floored
Logo	<syntaxhighlight lang="text" class="" style="" inline="1">REMAINDER</syntaxhighlight>	Template:Yes	Template:No	Truncated
Lua 5	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>	Template:Yes	Template:Yes	Floored
Lua 4	<syntaxhighlight lang="text" class="" style="" inline="1">mod(x,y)</syntaxhighlight>	Template:Yes	Template:Yes	Truncated
Liberty BASIC	<syntaxhighlight lang="text" class="" style="" inline="1">MOD</syntaxhighlight>	Template:Yes	Template:No	Truncated
Mathcad	<syntaxhighlight lang="text" class="" style="" inline="1">mod(x,y)</syntaxhighlight>	Template:Yes	Template:No	Floored
Maple	<syntaxhighlight lang="text" class="" style="" inline="1">e mod m</syntaxhighlight> (by default), <syntaxhighlight lang="text" class="" style="" inline="1">modp(e, m)</syntaxhighlight>	Template:Yes	Template:No	Euclidean
	<syntaxhighlight lang="text" class="" style="" inline="1">mods(e, m)</syntaxhighlight>	Template:Yes	Template:No	Rounded
	<syntaxhighlight lang="text" class="" style="" inline="1">frem(e, m)</syntaxhighlight>	Template:Yes	Template:Yes	Rounded
Mathematica	<syntaxhighlight lang="text" class="" style="" inline="1">Mod[a, b]</syntaxhighlight>	Template:Yes	Template:No	Floored
MATLAB	<syntaxhighlight lang="text" class="" style="" inline="1">mod</syntaxhighlight>	Template:Yes	Template:Yes	Floored
MATLAB	<syntaxhighlight lang="text" class="" style="" inline="1">rem</syntaxhighlight>	Template:Yes	Template:Yes	Truncated
Maxima	<syntaxhighlight lang="text" class="" style="" inline="1">mod</syntaxhighlight>	Template:Yes	Template:No	Floored
Maxima	<syntaxhighlight lang="text" class="" style="" inline="1">remainder</syntaxhighlight>	Template:Yes	Template:No	Truncated
Maya Embedded Language	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>	Template:Yes	Template:No	Truncated
Microsoft Excel	<syntaxhighlight lang="text" class="" style="" inline="1">=MOD()</syntaxhighlight>	Template:Yes	Template:Yes	Floored
Minitab	<syntaxhighlight lang="text" class="" style="" inline="1">MOD</syntaxhighlight>	Template:Yes	Template:No	Floored
Modula-2	<syntaxhighlight lang="text" class="" style="" inline="1">MOD</syntaxhighlight>	Template:Yes	Template:No	Floored
Modula-2	<syntaxhighlight lang="text" class="" style="" inline="1">REM</syntaxhighlight>	Template:Yes	Template:No	Truncated
MUMPS	<syntaxhighlight lang="text" class="" style="" inline="1">#</syntaxhighlight>	Template:Yes	Template:No	Floored
Netwide Assembler (NASM, NASMX)	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>, <syntaxhighlight lang="text" class="" style="" inline="1">div</syntaxhighlight> (unsigned)	Template:Yes	Template:No	Template:N/A
Netwide Assembler (NASM, NASMX)	<syntaxhighlight lang="text" class="" style="" inline="1">%%</syntaxhighlight> (signed)	Template:Yes	Template:No	citation	CitationClass=web }}</ref>
Nim	<syntaxhighlight lang="text" class="" style="" inline="1">mod</syntaxhighlight>	Template:Yes	Template:No	Truncated
Oberon	<syntaxhighlight lang="text" class="" style="" inline="1">MOD</syntaxhighlight>	Template:Yes	Template:No	Floored-likeTemplate:Efn
Objective-C	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>	Template:Yes	Template:No	Truncated (same as C99)
Object Pascal, Delphi	<syntaxhighlight lang="text" class="" style="" inline="1">mod</syntaxhighlight>	Template:Yes	Template:No	Truncated
OCaml	<syntaxhighlight lang="text" class="" style="" inline="1">mod</syntaxhighlight>	Template:Yes	Template:No	citation	CitationClass=web }}</ref>
OCaml	<syntaxhighlight lang="text" class="" style="" inline="1">mod_float</syntaxhighlight>	Template:No	Template:Yes	citation	CitationClass=web }}</ref>
Occam	<syntaxhighlight lang="text" class="" style="" inline="1">\</syntaxhighlight>	Template:Yes	Template:No	Truncated
Pascal (ISO-7185 and -10206)	<syntaxhighlight lang="text" class="" style="" inline="1">mod</syntaxhighlight>	Template:Yes	Template:No	Euclidean-likeTemplate:Efn
Perl	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>	Template:Yes	Template:No	FlooredTemplate:Efn
Perl	<syntaxhighlight lang="text" class="" style="" inline="1">POSIX::fmod</syntaxhighlight>	Template:No	Template:Yes	Truncated
Phix	<syntaxhighlight lang="text" class="" style="" inline="1">mod</syntaxhighlight>	Template:Yes	Template:No	Floored
Phix	<syntaxhighlight lang="text" class="" style="" inline="1">remainder</syntaxhighlight>	Template:Yes	Template:No	Truncated
PHP	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>	Template:Yes	Template:No	citation	CitationClass=web }}</ref>
PHP	<syntaxhighlight lang="text" class="" style="" inline="1">fmod</syntaxhighlight>	Template:No	Template:Yes	citation	CitationClass=web }}</ref>
PIC BASIC Pro	<syntaxhighlight lang="text" class="" style="" inline="1">\\</syntaxhighlight>	Template:Yes	Template:No	Truncated
PL/I	<syntaxhighlight lang="text" class="" style="" inline="1">mod</syntaxhighlight>	Template:Yes	Template:No	Floored (ANSI PL/I)
PowerShell	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>	Template:Yes	Template:No	Truncated
Programming Code (PRC)	<syntaxhighlight lang="text" class="" style="" inline="1">MATH.OP - 'MOD; (\)'</syntaxhighlight>	Template:Yes	Template:No	Undefined
Progress	<syntaxhighlight lang="text" class="" style="" inline="1">modulo</syntaxhighlight>	Template:Yes	Template:No	Truncated
Prolog (ISO 1995)	<syntaxhighlight lang="text" class="" style="" inline="1">mod</syntaxhighlight>	Template:Yes	Template:No	Floored
Prolog (ISO 1995)	<syntaxhighlight lang="text" class="" style="" inline="1">rem</syntaxhighlight>	Template:Yes	Template:No	Truncated
PureBasic	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>, <syntaxhighlight lang="text" class="" style="" inline="1">Mod(x,y)</syntaxhighlight>	Template:Yes	Template:No	Truncated
PureScript	<syntaxhighlight lang="text" class="" style="" inline="1">`mod`</syntaxhighlight>	Template:Yes	Template:No	citation	CitationClass=web }}</ref>
Pure Data	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>	Template:Yes	Template:No	Truncated (same as C)
Pure Data	<syntaxhighlight lang="text" class="" style="" inline="1">mod</syntaxhighlight>	Template:Yes	Template:No	Floored
Python	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>	Template:Yes	Template:Yes	Floored
	<syntaxhighlight lang="text" class="" style="" inline="1">math.fmod</syntaxhighlight>	Template:No	Template:Yes	Truncated
	<syntaxhighlight lang="text" class="" style="" inline="1">math.remainder</syntaxhighlight>	Template:No	Template:Yes	Rounded
Q#	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>	Template:Yes	Template:No	citation	CitationClass=web }}</ref>
R	<syntaxhighlight lang="text" class="" style="" inline="1">%%</syntaxhighlight>	Template:Yes	Template:Yes	citation	CitationClass=web }}</ref>
Racket	<syntaxhighlight lang="text" class="" style="" inline="1">modulo</syntaxhighlight>	Template:Yes	Template:No	Floored
Racket	<syntaxhighlight lang="text" class="" style="" inline="1">remainder</syntaxhighlight>	Template:Yes	Template:No	Truncated
Raku	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>	Template:No	Template:Yes	Floored
RealBasic	<syntaxhighlight lang="text" class="" style="" inline="1">MOD</syntaxhighlight>	Template:Yes	Template:No	Truncated
Reason	<syntaxhighlight lang="text" class="" style="" inline="1">mod</syntaxhighlight>	Template:Yes	Template:No	Truncated
Rexx	<syntaxhighlight lang="text" class="" style="" inline="1">//</syntaxhighlight>	Template:Yes	Template:Yes	Truncated
RPG	<syntaxhighlight lang="text" class="" style="" inline="1">%REM</syntaxhighlight>	Template:Yes	Template:No	Truncated
Ruby	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>, <syntaxhighlight lang="text" class="" style="" inline="1">modulo()</syntaxhighlight>	Template:Yes	Template:Yes	Floored
Ruby	<syntaxhighlight lang="text" class="" style="" inline="1">remainder()</syntaxhighlight>	Template:Yes	Template:Yes	Truncated
Rust	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>	Template:Yes	Template:Yes	Truncated
Rust	<syntaxhighlight lang="text" class="" style="" inline="1">rem_euclid()</syntaxhighlight>	Template:Yes	Template:Yes	citation	CitationClass=web }}</ref>
SAS	<syntaxhighlight lang="text" class="" style="" inline="1">MOD</syntaxhighlight>	Template:Yes	Template:No	Truncated
Scala	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>	Template:Yes	Template:Yes	Truncated
Scheme	<syntaxhighlight lang="text" class="" style="" inline="1">modulo</syntaxhighlight>	Template:Yes	Template:No	Floored
Scheme	<syntaxhighlight lang="text" class="" style="" inline="1">remainder</syntaxhighlight>	Template:Yes	Template:No	Truncated
Scheme R⁶RS	<syntaxhighlight lang="text" class="" style="" inline="1">mod</syntaxhighlight>	Template:Yes	Template:No	Euclidean<ref name="r6rs">r6rs.org</ref>
	<syntaxhighlight lang="text" class="" style="" inline="1">mod0</syntaxhighlight>	Template:Yes	Template:No	Rounded<ref name="r6rs"/>
	<syntaxhighlight lang="text" class="" style="" inline="1">flmod</syntaxhighlight>	Template:No	Template:Yes	Euclidean
	<syntaxhighlight lang="text" class="" style="" inline="1">flmod0</syntaxhighlight>	Template:No	Template:Yes	Rounded
Scratch	<syntaxhighlight lang="text" class="" style="" inline="1">mod</syntaxhighlight>	Template:Yes	Template:Yes	Floored
Seed7	<syntaxhighlight lang="text" class="" style="" inline="1">mod</syntaxhighlight>	Template:Yes	Template:Yes	Floored
Seed7	<syntaxhighlight lang="text" class="" style="" inline="1">rem</syntaxhighlight>	Template:Yes	Template:Yes	Truncated
SenseTalk	<syntaxhighlight lang="text" class="" style="" inline="1">modulo</syntaxhighlight>	Template:Yes	Template:No	Floored
SenseTalk	<syntaxhighlight lang="text" class="" style="" inline="1">rem</syntaxhighlight>	Template:Yes	Template:No	Truncated
<syntaxhighlight lang="text" class="" style="" inline="1">sh</syntaxhighlight> (POSIX) (includes bash, mksh, &c.)	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>	Template:Yes	Template:No	citation	CitationClass=web }}</ref>
Smalltalk	<syntaxhighlight lang="text" class="" style="" inline="1">\\</syntaxhighlight>	Template:Yes	Template:No	Floored
Smalltalk	<syntaxhighlight lang="text" class="" style="" inline="1">rem:</syntaxhighlight>	Template:Yes	Template:No	Truncated
Snap!	<syntaxhighlight lang="text" class="" style="" inline="1">mod</syntaxhighlight>	Template:Yes	Template:No	Floored
Spin	<syntaxhighlight lang="text" class="" style="" inline="1">//</syntaxhighlight>	Template:Yes	Template:No	Floored
Solidity	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>	Template:Yes	Template:No	citation	CitationClass=web }}</ref>
SQL (SQL:1999)	<syntaxhighlight lang="text" class="" style="" inline="1">mod(x,y)</syntaxhighlight>	Template:Yes	Template:No	Truncated
SQL (SQL:2011)	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>	Template:Yes	Template:No	Truncated
Standard ML	<syntaxhighlight lang="text" class="" style="" inline="1">mod</syntaxhighlight>	Template:Yes	Template:No	Floored
	<syntaxhighlight lang="text" class="" style="" inline="1">Int.rem</syntaxhighlight>	Template:Yes	Template:No	Truncated
	<syntaxhighlight lang="text" class="" style="" inline="1">Real.rem</syntaxhighlight>	Template:No	Template:Yes	Truncated
Stata	<syntaxhighlight lang="text" class="" style="" inline="1">mod(x,y)</syntaxhighlight>	Template:Yes	Template:No	Euclidean
Swift	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>	Template:Yes	Template:No	citation	CitationClass=web }}</ref>
	<syntaxhighlight lang="text" class="" style="" inline="1">remainder(dividingBy:)</syntaxhighlight>	Template:No	Template:Yes	citation	CitationClass=web }}</ref>
	<syntaxhighlight lang="text" class="" style="" inline="1">truncatingRemainder(dividingBy:)</syntaxhighlight>	Template:No	Template:Yes	citation	CitationClass=web }}</ref>
Tcl	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>	Template:Yes	Template:No	Floored
Tcl	<syntaxhighlight lang="text" class="" style="" inline="1">fmod()</syntaxhighlight>	Template:No	Template:Yes	Truncated (as C)
tcsh	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>	Template:Yes	Template:No	Truncated
Torque	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>	Template:Yes	Template:No	Truncated
Turing	<syntaxhighlight lang="text" class="" style="" inline="1">mod</syntaxhighlight>	Template:Yes	Template:No	Floored
Verilog (2001)	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>	Template:Yes	Template:No	Truncated
VHDL	<syntaxhighlight lang="text" class="" style="" inline="1">mod</syntaxhighlight>	Template:Yes	Template:No	Floored
VHDL	<syntaxhighlight lang="text" class="" style="" inline="1">rem</syntaxhighlight>	Template:Yes	Template:No	Truncated
VimL	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>	Template:Yes	Template:No	Truncated
Visual Basic	<syntaxhighlight lang="text" class="" style="" inline="1">Mod</syntaxhighlight>	Template:Yes	Template:No	Truncated
WebAssembly	<syntaxhighlight lang="text" class="" style="" inline="1">i32.rem_u</syntaxhighlight>, <syntaxhighlight lang="text" class="" style="" inline="1">i64.rem_u</syntaxhighlight> (unsigned)	Template:Yes	Template:No	Template:N/A<ref name="wasmcs2">{{#invoke:citation/CS1\|citation	CitationClass=web }}</ref>
WebAssembly	<syntaxhighlight lang="text" class="" style="" inline="1">i32.rem_s</syntaxhighlight>, <syntaxhighlight lang="text" class="" style="" inline="1">i64.rem_s</syntaxhighlight> (signed)	Template:Yes	Template:No	Truncated<ref name="wasmcs2" />
x86 assembly	<syntaxhighlight lang="text" class="" style="" inline="1">IDIV</syntaxhighlight>	Template:Yes	Template:No	Truncated
XBase++	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>	Template:Yes	Template:Yes	Truncated
XBase++	<syntaxhighlight lang="text" class="" style="" inline="1">Mod()</syntaxhighlight>	Template:Yes	Template:Yes	Floored
Zig	<syntaxhighlight lang="text" class="" style="" inline="1">%</syntaxhighlight>, <syntaxhighlight lang="text" class="" style="" inline="1">@rem</syntaxhighlight>	Template:Yes	Template:Yes	citation	CitationClass=web }}</ref>
Zig	<syntaxhighlight lang="text" class="" style="" inline="1">@mod</syntaxhighlight>	Template:Yes	Template:Yes	Floored
Z3 theorem prover	<syntaxhighlight lang="text" class="" style="" inline="1">div</syntaxhighlight>, <syntaxhighlight lang="text" class="" style="" inline="1">mod</syntaxhighlight>	Template:Yes	Template:No	Euclidean

In addition, many computer systems provide a <syntaxhighlight lang="text" class="" style="" inline="1">divmod</syntaxhighlight> functionality, which produces the quotient and the remainder at the same time. Examples include the x86 architecture's <syntaxhighlight lang="text" class="" style="" inline="1">IDIV</syntaxhighlight> instruction, the C programming language's <syntaxhighlight lang="text" class="" style="" inline="1">div()</syntaxhighlight> function, and Python's <syntaxhighlight lang="text" class="" style="" inline="1">divmod()</syntaxhighlight> function.

GeneralizationsEdit

Modulo with offsetEdit

Sometimes it is useful for the result of Template:Mvar modulo Template:Mvar to lie not between 0 and Template:Math, but between some number Template:Mvar and Template:Math. In that case, Template:Mvar is called an offset and Template:Math is particularly common.

There does not seem to be a standard notation for this operation, so let us tentatively use Template:Math. We thus have the following definition:<ref name="Mathematica Mod" >{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Template:Math just in case Template:Math and Template:Math. Clearly, the usual modulo operation corresponds to zero offset: Template:Math.

The operation of modulo with offset is related to the floor function as follows:

<math>a \operatorname{mod}_d n = a - n \left\lfloor\frac{a-d}{n}\right\rfloor.</math>

To see this, let <math display="inline">x = a - n \left\lfloor\frac{a-d}{n}\right\rfloor</math>. We first show that Template:Math. It is in general true that Template:Math for all integers Template:Mvar; thus, this is true also in the particular case when <math display="inline">b = -\!\left\lfloor\frac{a-d}{n}\right\rfloor</math>; but that means that <math display="inline">x \bmod n = \left(a - n \left\lfloor\frac{a-d}{n}\right\rfloor\right)\! \bmod n = a \bmod n</math>, which is what we wanted to prove. It remains to be shown that Template:Math. Let Template:Mvar and Template:Mvar be the integers such that Template:Math with Template:Math (see Euclidean division). Then <math display="inline">\left\lfloor\frac{a-d}{n}\right\rfloor = k</math>, thus <math display="inline">x = a - n \left\lfloor\frac{a-d}{n}\right\rfloor = a - n k = d +r</math>. Now take Template:Math and add Template:Mvar to both sides, obtaining Template:Math. But we've seen that Template:Math, so we are done.

The modulo with offset Template:Math is implemented in Mathematica as <syntaxhighlight lang="text" class="" style="" inline="1">Mod[a, n, d]</syntaxhighlight> .<ref name="Mathematica Mod" />

Implementing other modulo definitions using truncationEdit

Despite the mathematical elegance of Knuth's floored division and Euclidean division, it is generally much more common to find a truncated division-based modulo in programming languages. Leijen provides the following algorithms for calculating the two divisions given a truncated integer division:<ref name="Leijen" />

<syntaxhighlight lang="c"> /* Euclidean and Floored divmod, in the style of C's ldiv() */ typedef struct {

 /* This structure is part of the C stdlib.h, but is reproduced here for clarity */
 long int quot;
 long int rem;

} ldiv_t;

/* Euclidean division */ inline ldiv_t ldivE(long numer, long denom) {

 /* The C99 and C++11 languages define both of these as truncating. */
 long q = numer / denom;
 long r = numer % denom;
 if (r < 0) {
   if (denom > 0) {
     q = q - 1;
     r = r + denom;
   } else {
     q = q + 1;
     r = r - denom;
   }
 }
 return (ldiv_t){.quot = q, .rem = r};

}

/* Floored division */ inline ldiv_t ldivF(long numer, long denom) {

 long q = numer / denom;
 long r = numer % denom;
 if ((r > 0 && denom < 0) || (r < 0 && denom > 0)) {
   q = q - 1;
   r = r + denom;
 }
 return (ldiv_t){.quot = q, .rem = r};

} </syntaxhighlight>

For both cases, the remainder can be calculated independently of the quotient, but not vice versa. The operations are combined here to save screen space, as the logical branches are the same.

NotesEdit

Template:Notelist

ReferencesEdit

Template:Reflist

External linksEdit

Different kinds of integer division
Modulorama, animation of a cyclic representation of multiplication tables (explanation in French)

de:Division mit Rest#Modulo

Modulo

Contents

Variants of the definitionEdit

NotationEdit

Common pitfallsEdit

Performance issuesEdit

Properties (identities)Edit

In programming languagesEdit

GeneralizationsEdit

Modulo with offsetEdit

Implementing other modulo definitions using truncationEdit

See alsoEdit

NotesEdit

ReferencesEdit

External linksEdit