Modulo

---

title: "Modulo" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["computer-arithmetic", "articles-with-example-c++-code", "operators-(programming)", "modular-arithmetic", "operations-on-numbers"] description: "Computational operation" topic_path: "general/computer-arithmetic" source: "https://en.wikipedia.org/wiki/Modulo" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Computational operation ::

::callout[type=note] the binary operation mod(a,n) ::

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 a and n, a modulo n (often abbreviated as a mod n) is the remainder of the Euclidean division of a by n, where a is the dividend and n is the divisor.

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 a and n both being integers, many computing systems now allow other types of numeric operands. The range of values for an integer modulo operation of n is 0 to n − 1. a mod 1 is always 0.

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

Variants of the definition

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). 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 q and the remainder r of a divided by n \neq 0 satisfy the following conditions: &q \in \mathbb{Z} \ &a = n q + r \quad (n \neq 0) \ &|r| \end{align}|}}

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 a or n. 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 n or a is negative (see the table under for details). Some systems leave a modulo 0 undefined, though others define it as a.

| [[File:Divmod truncated.svg|thumb|upright=1.2| Quotient (q) and remainder (r) as functions of dividend (a), using truncated division]] Many implementations use truncated division, for which the quotient is defined by q = \operatorname{trunc}\left(\frac{a}{n}\right) where \operatorname{trunc} is the integral part function (rounding toward zero), i.e. the truncation to zero significant digits. Thus according to equation (), the remainder has the same sign as the dividend a so can take values: r = a - n \operatorname{trunc}\left(\frac{a}{n}\right) | [[File:Divmod floored.svg|thumb|upright=1.2|Quotient and remainder using floored division]] Donald Knuth promotes floored division, for which the quotient is defined by q = \left\lfloor\frac{a}{n}\right\rfloor where \lfloor,\rfloor is the floor function (rounding down). Thus according to equation (), the remainder has the same sign as the divisor n: r = a - n \left\lfloor\frac{a}{n}\right\rfloor |[[File:Divmod Euclidean.svg|thumb|upright=1.2|Quotient and remainder using Euclidean division]] Raymond T. Boute promotes Euclidean division, for which the non-negative remainder r \in {0, 1, 2...} is defined by r := a - nq \ \mathrm{such\ that} \ {\color{red}{0 \leq r}} (Emphasis added.) Under this definition, we can say the following about the quotient q: \begin{align} q &= \frac{a - r}{n} \in \mathbb{Z} \ &= \text{sgn}(n) \cdot \frac{a-r}{|n|} \ &= \text{sgn}(n) \cdot \left( \frac{a}{|n|} - \frac{r}{|n|} \right) \ &= \text{sgn}(n) \cdot \left\lfloor \frac{a}{\left|n\right|} \right\rfloor \end{align} where sgn is the sign function, \lfloor,\rfloor is the floor function (rounding down), and \frac{a}{|n|} \in \mathbb{Q}, \frac{r}{|n|} \in \mathbb{Q} are rational numbers. Equivalently, one may instead define the quotient q \in \mathbb{Z} as follows: q := \sgn(n) \left\lfloor\frac{a}{\left|n\right|}\right\rfloor = \begin{cases} \left\lfloor\frac{a}{n}\right\rfloor & \text{if } n 0 \ \left\lceil\frac{a}{n}\right\rceil & \text{if } n \end{cases} where \lceil,\rceil is the ceiling function (rounding up). Thus according to equation (), the remainder r is non-negative: r = a - nq = a - |n| \left\lfloor\frac{a}{\left|n\right|}\right\rfloor | [[File:Divmod rounding.svg|thumb|upright=1.2|Quotient and remainder using rounded division]] Common Lisp and IEEE 754 use rounded division, for which the quotient is defined by q = \operatorname{round}\left(\frac{a}{n}\right) where round is the round function (rounding half to even). Thus according to equation (), the remainder falls between -\frac{n}{2} and \frac{n}{2}, and its sign depends on which side of zero it falls to be within these boundaries: r = a - n \operatorname{round}\left(\frac{a}{n}\right) | [[File:Divmod ceiling.svg|thumb|upright=1.2|Quotient and remainder using ceiling division]] Common Lisp also uses ceiling division, for which the quotient is defined by q = \left\lceil\frac{a}{n}\right\rceil where ⌈⌉ is the ceiling function (rounding up). Thus according to equation (), the remainder has the opposite sign of that of the divisor: r = a - n \left\lceil\frac{a}{n}\right\rceil

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.

However, truncated division satisfies the identity ({-a})/b = {-(a/b)} = a/({-b}).

Notation

::callout[type=note] the binary mod operation ::

Some calculators have a mod() function button, and many programming languages have a similar function, expressed as mod(a, n), for example. Some also support expressions that use "%", "mod", or "Mod" as a modulo or remainder operator, such as or .

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

Common pitfalls

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:

::code[lang=cpp] bool is_odd(int n) { return n % 2 == 1; } ::

But in a language where modulo has the sign of the dividend, that is incorrect, because when n (the dividend) is negative and odd, n 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):

::code[lang=cpp] bool is_odd(int n) { return n % 2 != 0; } ::

Or with the binary arithmetic: ::code[lang=cpp] bool is_odd(int n) { return n & 1; } ::

Performance issues

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 x 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.

Compiler optimizations may recognize expressions of the form where is a power of two and automatically implement them as , 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 will always be positive. For these languages, the equivalence x % 2n == x n - 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)

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 a and n are integers.

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

In programming languages

::data[format=table title="Modulo operators in various programming languages"]

Language	Operator	Integer	Floating-point	Definition
ABAP				Euclidean
ActionScript				Truncated
Ada				title=ISO/IEC 8652:2012 - Information technology — Programming languages — Ada
			Truncated
ALGOL 68	, , , , **mod**			Euclidean
AMPL				Truncated
APL	`	`
AppleScript				Truncated
AutoLISP				Truncated
AWK				Truncated (same as in C)
BASIC				Varies by implementation
bc				Truncated
CC++	,			Truncated
(C) (C++)			Truncated
(C) (C++)			Rounded
C#				Truncated
			last=dotnet-bot	title=Math.IEEERemainder(Double, Double) Method (System)
Clarion				Truncated
Clean				Truncated
Clojure				Floored
			Truncated
COBOL				title=ISO/IEC 1989:2023 – Programming language COBOL
			Truncated
CoffeeScript				Truncated
			Floored
ColdFusion	,			Truncated
Common Intermediate Language	(signed)			Truncated
(unsigned)
Common Lisp				title=CLHS: Function MOD, REM
			Truncated
Crystal	,			Floored
			Truncated
CSS				Floored
			Truncated
D				Truncated
Dart				Euclidean
			Truncated
Eiffel				Truncated
Elixir				Truncated
			Floored
Elm				Floored
			Truncated
Erlang				Truncated
			Truncated (same as C)
Euphoria				Truncated
			Floored
F#				Truncated (same as C#)
			Rounded
Factor				Truncated
			Euclidean
FileMaker				Floored
Forth				Implementation defined
			Floored
			Truncated
Fortran				Truncated
			Floored
Frink				Floored
Full BASIC				Floored
			Truncated
GLSL				Undefined
			Floored
GameMaker Studio (GML)	,			Truncated
GDScript (Godot)				Truncated
			Euclidean
			Truncated
			Euclidean
Go				Truncated
			Truncated
			Euclidean
			Truncated
Groovy				Truncated
Haskell				title=6 Predefined Types and Classes
			Truncated
(GHC)			Floored
Haxe				Truncated
HLSL				Undefined
J	`	`
Java				Truncated
			Floored
JavaScriptTypeScript				Truncated
Julia				Floored
,			Truncated
Kotlin	,			Truncated
			Floored
ksh				Truncated (same as POSIX )
			Truncated
LabVIEW				Truncated
LibreOffice				Floored
Logo				Floored
			Truncated
Lua 5				Floored
Lua 4				Truncated
Liberty BASIC				Truncated
Mathcad				Floored
Maple	(by default),			Euclidean
			Rounded
			Rounded
Mathematica				Floored
MATLAB				Floored
			Truncated
Maxima				Floored
			Truncated
Maya Embedded Language				Truncated
Microsoft Excel				Floored
Minitab				Floored
Modula-2				Floored
			Truncated
MUMPS				Floored
Netwide Assembler (NASM, NASMX)	, (unsigned)
(signed)			Implementation-defined
Nim				Truncated
Oberon				Floored-like
Objective-C				Truncated (same as C99)
Object Pascal, Delphi				Truncated
OCaml				Truncated
			Truncated
Occam				Truncated
Pascal (ISO-7185 and -10206)				Euclidean-like
Perl				Floored
			Truncated
PHP				Truncated
			Truncated
PIC BASIC Pro				Truncated
PL/I				Floored (ANSI PL/I)
PowerShell				Truncated
Programming Code (PRC)				Undefined
Progress				Truncated
Prolog (ISO 1995)				Floored
			Truncated
PureBasic	,			Truncated
PureScript				Euclidean
Pure Data				Truncated (same as C)
			Floored
Python				Floored
			Truncated
			Rounded
Q#				Truncated
R				Floored
Racket				Floored
			Truncated
Raku				Floored
RealBasic				Truncated
Reason				Truncated
Rexx				Truncated
RPG				Truncated
Ruby	,			Floored
			Truncated
Rust				Truncated
			Euclidean
SAS				Truncated
Scala				Truncated
Scheme				Floored
			Truncated
Scheme R6RS				Euclidean
			Rounded
			Euclidean
			Rounded
Scratch				Floored
Seed7				Floored
			Truncated
SenseTalk				Floored
			Truncated
(POSIX) (includes bash, mksh, &c.)				Truncated (same as C)
Smalltalk				title=ANSI INCITS 319-1998 (R2002): Information Technology - Programming Languages - Smalltalk
			at=sec. 5.6.2.30}}
Snap!				Floored
Spin				Floored
Solidity				Truncated
SQL (SQL:1999)				Truncated
SQL (SQL:2011)				Truncated
Standard ML				Floored
			Truncated
			Truncated
Stata				Euclidean
Swift				Truncated
			Rounded
			Truncated
Tcl				Floored
			Truncated (same as C)
tcsh				Truncated
Torque				Truncated
Turing				Floored
Verilog (2001)				Truncated
VHDL				Floored
			Truncated
VimL				Truncated
Visual Basic				Truncated
WebAssembly	, (unsigned)			editor-first1=Andreas
, (signed)			Truncated
x86 assembly				Truncated
XBase++				Truncated
			Floored
Zig	,			Truncated
			Floored
Z3 theorem prover	,			Euclidean
::

In addition, many computer systems provide a functionality, which produces the quotient and the remainder at the same time. Examples include the x86 architecture's instruction, the C programming language's function, and Python's function.

Generalizations

Modulo with offset

Sometimes it is useful for the result of a modulo n to lie not between 0 and n − 1, but between some number d and d + n − 1. In that case, d is called an offset and is particularly common.

There does not seem to be a standard notation for this operation, so let us tentatively use a modd n. We thus have the following definition: just in case d ≤ x ≤ d + n − 1 and . Clearly, the usual modulo operation corresponds to zero offset: .

The operation of modulo with offset is related to the floor function as follows: a \operatorname{mod}_d n = a - n \left\lfloor\frac{a-d}{n}\right\rfloor.

To see this, let x = a - n \left\lfloor\frac{a-d}{n}\right\rfloor. We first show that . It is in general true that for all integers b; thus, this is true also in the particular case when b = -!\left\lfloor\frac{a-d}{n}\right\rfloor; but that means that x \bmod n = \left(a - n \left\lfloor\frac{a-d}{n}\right\rfloor\right)! \bmod n = a \bmod n, which is what we wanted to prove. It remains to be shown that d ≤ x ≤ d + n − 1. Let k and r be the integers such that with 0 ≤ r ≤ n − 1 (see Euclidean division). Then \left\lfloor\frac{a-d}{n}\right\rfloor = k, thus x = a - n \left\lfloor\frac{a-d}{n}\right\rfloor = a - n k = d +r. Now take 0 ≤ r ≤ n − 1 and add d to both sides, obtaining d ≤ d + r ≤ d + n − 1. But we've seen that , so we are done.

The modulo with offset a modd n is implemented in Mathematica as  .

Implementing other modulo definitions using truncation

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:

::code[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}; } ::

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.

The other modes of rounding are: ::code[lang=c] /* Round-division */ ldiv_t ldivR(long numer, long denom) {

}

/* Ceiling-division */ ldiv_t ldivC(long numer, long denom) {

} ::

--

Notes

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