Modulo operation
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the modulus of the operation).
Given two positive numbers a and n, a modulo n (often abbreviated as a mod n or as a % n) is the remainder of the Euclidean division of a by n, where a is the dividend and n is the divisor. The modulo operation is to be distinguished from the symbol mod, which refers to the modulus^{[1]} (or divisor) one is operating from.
For example, the expression "5 mod 2" would evaluate 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 the division of 9 by 3 has a quotient of 3 and a remainder of 0; there is nothing to subtract from 9 after multiplying 3 times 3.
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 inclusive (a mod 1 is always 0; a mod 0 is undefined, possibly resulting in a division by zero error in some programming languages). See Modular arithmetic for an older and related convention applied in number theory.
When exactly one of a or n is negative, the naive 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 nonnegative integer that belongs to that class (i.e., the remainder of the Euclidean division).^{[2]} 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 satisfy the following conditions:
[math]\displaystyle{ \begin{align} q \,&\in \mathbb{Z} \\ a \,&= n q + r \\ r &\lt n \end{align} }[/math]
(
)
However, this still leaves a sign ambiguity if the remainder is nonzero: two possible choices for the remainder occur, one negative and the other positive, and two possible choices for the quotient occur. 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.^{[loweralpha 1]} 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 § In programming languages for details). a modulo 0 is undefined in most systems, although some do define it as a.
 Many implementations use truncated division, where the quotient is defined by truncation (integer part) [math]\displaystyle{ q = \operatorname{trunc}\left(\frac{a}{n}\right) }[/math] and thus according to equation (1) the remainder would have the same sign as the dividend. The quotient is rounded towards zero: equal to the first integer in the direction of zero from the exact rational quotient.
 [math]\displaystyle{ r = a  n \operatorname{trunc}\left(\frac{a}{n}\right) }[/math]
 Donald Knuth^{[3]} described floored division where the quotient is defined by the floor function [math]\displaystyle{ q = \left\lfloor\frac{a}{n}\right\rfloor }[/math] and thus according to equation (1) the remainder would have the same sign as the divisor. Due to the floor function, the quotient is always rounded downwards, even if it is already negative.
 [math]\displaystyle{ r = a  n \left\lfloor\frac{a}{n}\right\rfloor }[/math]
 Raymond T. Boute^{[4]} describes the Euclidean definition in which the remainder is nonnegative always, 0 ≤ r, and is thus consistent with the Euclidean division algorithm. In this case,
 [math]\displaystyle{ n \gt 0 \implies q = \left\lfloor\frac{a}{n}\right\rfloor }[/math]
 [math]\displaystyle{ n \lt 0 \implies q = \left\lceil\frac{a}{n}\right\rceil }[/math]
or equivalently
 [math]\displaystyle{ q = \sgn(n) \left\lfloor \frac{a}{\leftn\right} \right\rfloor }[/math]
where sgn is the sign function, and thus
 [math]\displaystyle{ r = a  n \left\lfloor \frac{a}{\leftn\right} \right\rfloor }[/math]
 A rounddivision is where the quotient is [math]\displaystyle{ q = \operatorname{round}\left(\frac{a}{n}\right) }[/math], i.e. rounded to the nearest integer. It is found in Common Lisp and IEEE 754 (see the round to nearest convention in IEEE754). Thus, the sign of the remainder is chosen to be nearest to zero.
 Common Lisp also defines ceilingdivision (remainder different sign from divisor) where the quotient is given by [math]\displaystyle{ q = \left\lceil\frac{a}{n}\right\rceil }[/math]. Thus, the sign of the remainder is chosen to be different from that of the divisor.
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.—Daan Leijen, Division and Modulus for Computer Scientists^{[5]}
However, truncated division satisfies the identity [math]\displaystyle{ ({a})/b = {(a/b)} = a/({b}) }[/math].^{[6]}
Notation
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 a % n
or a mod n
.
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 (truncating 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:
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):
bool is_odd(int n) { return n % 2 != 0; }
Another alternative is to use the fact that for any odd number, the remainder may be either 1 or −1:
bool is_odd(int n) { return n % 2 == 1  n % 2 == 1; }
A simpler alternative is to treat the result of n % 2 as if it is a boolean value, where any nonzero value is true:
bool is_odd(int n) { return n % 2; }
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 nontruncating definition):
x % 2^{n} == x & (2^{n}  1)
Examples:
x % 2 == x & 1
x % 4 == x & 3
x % 8 == x & 7
In devices and software that implement bitwise operations more efficiently than modulo, these alternative forms can result in faster calculations.^{[7]}
Compiler optimizations may recognize expressions of the form expression % constant
where constant
is a power of two and automatically implement them as expression & (constant1)
, 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 expression & (constant1)
will always be positive. For these languages, the equivalence x % 2^{n} == x < 0 ? x  ~(2^{n}  1) : x & (2^{n}  1)
has to be used instead, expressed using bitwise OR, NOT and AND operations.
Optimizations for general constantmodulus operations also exist by calculating the division first using the constantdivisor 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. Some of these properties require that a and n are integers.
 Identity:
 (a mod n) mod n = a mod n.
 n^{x} mod n = 0 for all positive integer values of x.
 If p is a prime number which is not a divisor of b, then ab^{p−1} mod p = a mod p, due to Fermat's little theorem.
 Inverse:
 [(−a mod n) + (a mod n)] mod n = 0.
 b^{−1} 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: [(b^{−1} mod n)(b mod n)] mod n = 1.
 Distributive:
 (a + b) mod n = [(a mod n) + (b mod n)] mod n.
 ab mod n = [(a mod n)(b mod n)] mod n.
 Division (definition): a/b mod n = [(a mod n)(b^{−1} mod n)] mod n, when the right hand side is defined (that is when b and n are coprime), and undefined otherwise.
 Inverse multiplication: [(ab mod n)(b^{−1} mod n)] mod n = a mod n.
In programming languages
Language  Operator  Integer  Floatingpoint  Definition 

ABAP  MOD

Yes  Yes  Euclidean 
ActionScript  %

Yes  No  Truncated 
Ada  mod

Yes  No  Floored^{[8]} 
rem

Yes  No  Truncated^{[8]}  
ALGOL 68  ÷× , mod

Yes  No  Euclidean 
AMPL  mod

Yes  No  Truncated 
APL   ^{[loweralpha 2]}

Yes  No  Floored 
AppleScript  mod

Yes  No  Truncated 
AutoLISP  (rem d n)

Yes  No  Truncated 
AWK  %

Yes  No  Truncated 
BASIC  Mod

Yes  No  Undefined 
bc  %

Yes  No  Truncated 
C C++ 
% , div

Yes  No  Truncated^{[loweralpha 3]} 
fmod (C)std::fmod (C++)

No  Yes  Truncated^{[11]}  
remainder (C)std::remainder (C++)

No  Yes  Rounded  
C#  %

Yes  Yes  Truncated 
Clarion  %

Yes  No  Truncated 
Clean  rem

Yes  No  Truncated 
Clojure  mod

Yes  No  Floored^{[12]} 
rem

Yes  No  Truncated^{[13]}  
COBOL  FUNCTION MOD

Yes  No  Floored^{[loweralpha 4]} 
CoffeeScript  %

Yes  No  Truncated 
%%

Yes  No  Floored^{[14]}  
ColdFusion  % , MOD

Yes  No  Truncated 
Common Lisp  mod

Yes  Yes  Floored 
rem

Yes  Yes  Truncated  
Crystal  % , modulo

Yes  Yes  Floored 
remainder

Yes  Yes  Truncated  
D  %

Yes  Yes  Truncated^{[15]} 
Dart  %

Yes  Yes  Euclidean^{[16]} 
remainder()

Yes  Yes  Truncated^{[17]}  
Eiffel  \\

Yes  No  Truncated 
Elixir  rem/2

Yes  No  Truncated^{[18]} 
Integer.mod/2

Yes  No  Floored^{[19]}  
Elm  modBy

Yes  No  Floored^{[20]} 
remainderBy

Yes  No  Truncated^{[21]}  
Erlang  rem

Yes  No  Truncated 
math:fmod/2

No  Yes  Truncated (same as C)^{[22]}  
Euphoria  mod

Yes  No  Floored 
remainder

Yes  No  Truncated  
F#  %

Yes  Yes  Truncated 
Factor  mod

Yes  No  Truncated 
FileMaker  Mod

Yes  No  Floored 
Forth  mod

Yes  No  Implementation defined 
fm/mod

Yes  No  Floored  
sm/rem

Yes  No  Truncated  
Fortran  mod

Yes  Yes  Truncated 
modulo

Yes  Yes  Floored  
Frink  mod

Yes  No  Floored 
GLSL  %

Yes  No  Undefined^{[23]} 
mod

No  Yes  Floored^{[24]}  
GameMaker Studio (GML)  mod , %

Yes  No  Truncated 
GDScript (Godot)  %

Yes  No  Truncated 
fmod

No  Yes  Truncated  
posmod

Yes  No  Floored  
fposmod

No  Yes  Floored  
Go  %

Yes  No  Truncated^{[25]} 
math.Mod

No  Yes  Truncated^{[26]}  
big.Int.Mod

Yes  No  Euclidean^{[27]}  
Groovy  %

Yes  No  Truncated 
Haskell  mod

Yes  No  Floored^{[28]} 
rem

Yes  No  Truncated^{[28]}  
Data.Fixed.mod' (GHC)

No  Yes  Floored  
Haxe  %

Yes  No  Truncated 
HLSL  %

Yes  Yes  Undefined^{[29]} 
J   ^{[loweralpha 2]}

Yes  No  Floored 
Java  %

Yes  Yes  Truncated 
Math.floorMod

Yes  No  Floored  
JavaScript TypeScript 
%

Yes  Yes  Truncated 
Julia  mod

Yes  Yes  Floored^{[30]} 
% , rem

Yes  Yes  Truncated^{[31]}  
Kotlin  % , rem

Yes  Yes  Truncated^{[32]} 
mod

Yes  Yes  Floored^{[33]}  
ksh  %

Yes  No  Truncated (same as POSIX sh) 
fmod

No  Yes  Truncated  
LabVIEW  mod

Yes  Yes  Truncated 
LibreOffice  =MOD()

Yes  No  Floored 
Logo  MODULO

Yes  No  Floored 
REMAINDER

Yes  No  Truncated  
Lua 5  %

Yes  Yes  Floored 
Lua 4  mod(x,y)

Yes  Yes  Truncated 
Liberty BASIC  MOD

Yes  No  Truncated 
Mathcad  mod(x,y)

Yes  No  Floored 
Maple  e mod m (by default), modp(e, m)

Yes  No  Euclidean 
mods(e, m)

Yes  No  Rounded  
frem(e, m)

Yes  Yes  Rounded  
Mathematica  Mod[a, b]

Yes  No  Floored 
MATLAB  mod

Yes  No  Floored 
rem

Yes  No  Truncated  
Maxima  mod

Yes  No  Floored 
remainder

Yes  No  Truncated  
Maya Embedded Language  %

Yes  No  Truncated 
Microsoft Excel  =MOD()

Yes  Yes  Floored 
Minitab  MOD

Yes  No  Floored 
Modula2  MOD

Yes  No  Floored 
REM

Yes  No  Truncated  
MUMPS  #

Yes  No  Floored 
Netwide Assembler (NASM, NASMX)  % , div (unsigned)

Yes  No  N/A 
%% (signed)

Yes  No  Implementationdefined^{[34]}  
Nim  mod

Yes  No  Truncated 
Oberon  MOD

Yes  No  Flooredlike^{[loweralpha 5]} 
ObjectiveC  %

Yes  No  Truncated (same as C99) 
Object Pascal, Delphi  mod

Yes  No  Truncated 
OCaml  mod

Yes  No  Truncated^{[35]} 
mod_float

No  Yes  Truncated^{[36]}  
Occam  \

Yes  No  Truncated 
Pascal (ISO7185 and 10206)  mod

Yes  No  Euclideanlike^{[loweralpha 6]} 
Programming Code Advanced (PCA)  \

Yes  No  Undefined 
Perl  %

Yes  No  Floored^{[loweralpha 7]} 
POSIX::fmod

No  Yes  Truncated  
Phix  mod

Yes  No  Floored 
remainder

Yes  No  Truncated  
PHP  %

Yes  No  Truncated^{[38]} 
fmod

No  Yes  Truncated^{[39]}  
PIC BASIC Pro  \\

Yes  No  Truncated 
PL/I  mod

Yes  No  Floored (ANSI PL/I) 
PowerShell  %

Yes  No  Truncated 
Programming Code (PRC)  MATH.OP  'MOD; (\)'

Yes  No  Undefined 
Progress  modulo

Yes  No  Truncated 
Prolog (ISO 1995)  mod

Yes  No  Floored 
rem

Yes  No  Truncated  
PureBasic  % , Mod(x,y)

Yes  No  Truncated 
PureScript  `mod`

Yes  No  Euclidean^{[40]} 
Pure Data  %

Yes  No  Truncated (same as C) 
mod

Yes  No  Floored  
Python  %

Yes  Yes  Floored 
math.fmod

No  Yes  Truncated  
Q#  %

Yes  No  Truncated^{[41]} 
R  %%

Yes  No  Floored 
Racket  modulo

Yes  No  Floored 
remainder

Yes  No  Truncated  
Raku  %

No  Yes  Floored 
RealBasic  MOD

Yes  No  Truncated 
Reason  mod

Yes  No  Truncated 
Rexx  //

Yes  Yes  Truncated 
RPG  %REM

Yes  No  Truncated 
Ruby  % , modulo()

Yes  Yes  Floored 
remainder()

Yes  Yes  Truncated  
Rust  %

Yes  Yes  Truncated 
rem_euclid()

Yes  Yes  Euclidean^{[42]}  
SAS  MOD

Yes  No  Truncated 
Scala  %

Yes  No  Truncated 
Scheme  modulo

Yes  No  Floored 
remainder

Yes  No  Truncated  
Scheme R^{6}RS  mod

Yes  No  Euclidean^{[43]} 
mod0

Yes  No  Rounded^{[43]}  
flmod

No  Yes  Euclidean  
flmod0

No  Yes  Rounded  
Scratch  mod

Yes  Yes  Floored 
Seed7  mod

Yes  Yes  Floored 
rem

Yes  Yes  Truncated  
SenseTalk  modulo

Yes  No  Floored 
rem

Yes  No  Truncated  
sh (POSIX) (includes bash, mksh, &c.)

%

Yes  No  Truncated (same as C)^{[44]} 
Smalltalk  \\

Yes  No  Floored 
rem:

Yes  No  Truncated  
Snap!  mod

Yes  No  Floored 
Spin  //

Yes  No  Floored 
Solidity  %

Yes  No  Floored 
SQL (1999)  mod(x,y)

Yes  No  Truncated 
SQL (2011)  %

Yes  No  Truncated 
Standard ML  mod

Yes  No  Floored 
Int.rem

Yes  No  Truncated  
Real.rem

No  Yes  Truncated  
Stata  mod(x,y)

Yes  No  Euclidean 
Swift  %

Yes  No  Truncated^{[45]} 
remainder(dividingBy:)

No  Yes  Rounded^{[46]}  
truncatingRemainder(dividingBy:)

No  Yes  Truncated^{[47]}  
Tcl  %

Yes  No  Floored 
Torque  %

Yes  No  Truncated 
Turing  mod

Yes  No  Floored 
Verilog (2001)  %

Yes  No  Truncated 
VHDL  mod

Yes  No  Floored 
rem

Yes  No  Truncated  
VimL  %

Yes  No  Truncated 
Visual Basic  Mod

Yes  No  Truncated 
WebAssembly  i32.rem_u , i64.rem_u (unsigned)

Yes  No  N/A^{[48]} 
i32.rem_s , i64.rem_s (signed)

Yes  No  Truncated^{[49]}  
x86 assembly  IDIV

Yes  No  Truncated 
XBase++  %

Yes  Yes  Truncated 
Mod()

Yes  Yes  Floored  
Z3 theorem prover  div , mod

Yes  No  Euclidean 
In addition, many computer systems provide a divmod
functionality, which produces the quotient and the remainder at the same time. Examples include the x86 architecture's IDIV
instruction, the C programming language's div()
function, and Python's divmod()
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. There does not seem to be a standard notation for this operation, so let us tentatively use a mod_{d} n. We thus have the following definition:^{[50]} x = a mod_{d} n just in case d ≤ x ≤ d + n − 1 and x mod n = a mod n. Clearly, the usual modulo operation corresponds to zero offset: a mod n = a mod_{0} n. The operation of modulo with offset is related to the floor function as follows:
 [math]\displaystyle{ a \operatorname{mod}_d n = a  n \left\lfloor\frac{ad}{n}\right\rfloor. }[/math]
(To see this, let [math]\displaystyle{ x = a  n \left\lfloor\frac{ad}{n}\right\rfloor }[/math]. We first show that x mod n = a mod n. It is in general true that (a + bn) mod n = a mod n for all integers b; thus, this is true also in the particular case when [math]\displaystyle{ b = \!\left\lfloor\frac{ad}{n}\right\rfloor }[/math]; but that means that [math]\displaystyle{ x \bmod n = \left(a  n \left\lfloor\frac{ad}{n}\right\rfloor\right)\! \bmod n = a \bmod n }[/math], 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 a − d = kn + r with 0 ≤ r ≤ n − 1 (see Euclidean division). Then [math]\displaystyle{ \left\lfloor\frac{ad}{n}\right\rfloor = k }[/math], thus [math]\displaystyle{ x = a  n \left\lfloor\frac{ad}{n}\right\rfloor = a  n k = d +r }[/math]. Now take 0 ≤ r ≤ n − 1 and add d to both sides, obtaining d ≤ d + r ≤ d + n − 1. But we've seen that x = d + r, so we are done. □)
The modulo with offset a mod_{d} n is implemented in Mathematica as Mod[a, n, d]
.^{[50]}
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 divisionbased modulo in programming languages. Leijen provides the following algorithms for calculating the two divisions given a truncated integer division:^{[5]}
/* 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.
See also
 Modulo (disambiguation) and modulo (jargon) – many uses of the word modulo, all of which grew out of Carl F. Gauss's introduction of modular arithmetic in 1801.
 Modulo (mathematics), general use of the term in mathematics
 Modular exponentiation
 Turn (unit)
Notes
 ↑ Mathematically, these two choices are but two of the infinite number of choices available for the inequality satisfied by a remainder.
 ↑ ^{2.0} ^{2.1} Argument order reverses, i.e.,
αω
computes [math]\displaystyle{ \omega\bmod\alpha }[/math], the remainder when dividingω
byα
.  ↑ C99 and C++11 define the behavior of
%
to be truncated.^{[9]} The standards before then leave the behavior implementationdefined.^{[10]}  ↑ As implemented in ACUCOBOL, Micro Focus COBOL, and possible others.
 ↑ Divisor must be positive, otherwise undefined.
 ↑ As discussed by Boute, ISO Pascal's definitions of
div
andmod
do not obey the Division Identity of D = d · (D / d) + D % d, and are thus fundamentally broken.  ↑ Perl usually uses arithmetic modulo operator that is machineindependent. For examples and exceptions, see the Perl documentation on multiplicative operators.^{[37]}
References
 ↑ Weisstein, Eric W.. "Congruence" (in en). https://mathworld.wolfram.com/Congruence.html.
 ↑ Caldwell, Chris. "residue". https://primes.utm.edu/glossary/page.php?sort=Residue.
 ↑ Knuth, Donald. E. (1972). The Art of Computer Programming. AddisonWesley. https://archive.org/details/artofcomputerpro0003knut.
 ↑ Boute, Raymond T. (April 1992). "The Euclidean definition of the functions div and mod". ACM Transactions on Programming Languages and Systems (ACM Press (New York, NY, USA)) 14 (2): 127–144. doi:10.1145/128861.128862. http://portal.acm.org/citation.cfm?id=128862&coll=portal&dl=ACM.
 ↑ ^{5.0} ^{5.1} Leijen, Daan (December 3, 2001). Division and Modulus for Computer Scientists. http://research.microsoft.com/pubs/151917/divmodnote.pdf. Retrieved 20141225.
 ↑ Peterson, Doctor (5 July 2001). "Mod Function and Negative Numbers". http://mathforum.org/library/drmath/view/52343.html.
 ↑ Horvath, Adam (July 5, 2012). "Faster division and modulo operation  the power of two". http://blog.teamleadnet.com/2012/07/fasterdivisionandmodulooperation.html.
 ↑ ^{8.0} ^{8.1} ISO/IEC 8652:2012  Information technology — Programming languages — Ada. ISO, IEC. 2012. sec. 4.5.5 Multiplying Operators.
 ↑ "C99 specification (ISO/IEC 9899:TC2)". 20050506. sec. 6.5.5 Multiplicative operators. http://www.openstd.org/jtc1/sc22/wg14/www/docs/n1124.pdf.
 ↑ ISO/IEC 14882:2003: Programming languages – C++. International Organization for Standardization (ISO), International Electrotechnical Commission (IEC). 2003. sec. 5.6.4. "the binary % operator yields the remainder from the division of the first expression by the second. .... If both operands are nonnegative then the remainder is nonnegative; if not, the sign of the remainder is implementationdefined".
 ↑ ISO/IEC 9899:1990: Programming languages – C. ISO, IEC. 1990. sec. 7.5.6.4. "The fmod function returns the value x  i * y, for some integer i such that, if y is nonzero, the result has the same sign as x and magnitude less than the magnitude of y.".
 ↑ "clojure.core  Clojure v1.10.3 API documentation". https://clojure.github.io/clojure/clojure.coreapi.html#clojure.core/mod.
 ↑ "clojure.core  Clojure v1.10.3 API documentation". https://clojure.github.io/clojure/clojure.coreapi.html#clojure.core/rem.
 ↑ CoffeeScript operators
 ↑ "Expressions  D Programming Language". https://dlang.org/spec/expression.html#mul_expressions.
 ↑ "operator % method  num class  dart:core library  Dart API". https://api.dart.dev/stable/2.13.1/dartcore/num/operator_modulo.html.
 ↑ "remainder method  num class  dart:core library  Dart API". https://api.dart.dev/stable/2.13.1/dartcore/num/remainder.html.
 ↑ "Kernel — Elixir v1.11.3". https://hexdocs.pm/elixir/Kernel.html#rem/2.
 ↑ "Integer — Elixir v1.11.3". https://hexdocs.pm/elixir/Integer.html#mod/2.
 ↑ "Basics  core 1.0.5". https://package.elmlang.org/packages/elm/core/latest/Basics#modBy.
 ↑ "Basics  core 1.0.5". https://package.elmlang.org/packages/elm/core/latest/Basics#remainderBy.
 ↑ "Erlang  math". https://erlang.org/doc/man/math.html#fmod2.
 ↑ "GLSL Language Specification, Version 4.50.7". section 5.9 Expressions. https://www.khronos.org/registry/OpenGL/specs/gl/GLSLangSpec.4.50.pdf. "If both operands are nonnegative, then the remainder is nonnegative. Results are undefined if one or both operands are negative."
 ↑ "GLSL Language Specification, Version 4.50.7". section 8.3 Common Functions. https://www.khronos.org/registry/OpenGL/specs/gl/GLSLangSpec.4.50.pdf.
 ↑ "The Go Programming Language Specification  The Go Programming Language". https://go.dev/ref/spec#Integer_operators.
 ↑ "math package  math  pkg.go.dev". https://pkg.go.dev/math#Mod.
 ↑ "big package  math/big  pkg.go.dev". https://pkg.go.dev/math/big#Int.Mod.
 ↑ ^{28.0} ^{28.1} "6 Predefined Types and Classes". https://www.haskell.org/onlinereport/haskell2010/haskellch6.html#x131370006.4.2.
 ↑ "Operators". Microsoft. https://docs.microsoft.com/enus/windows/win32/direct3dhlsl/dxgraphicshlsloperators#additiveandmultiplicativeoperators. "The % operator is defined only in cases where either both sides are positive or both sides are negative. Unlike C, it also operates on floatingpoint data types, as well as integers."
 ↑ "Mathematics · The Julia Language". https://docs.julialang.org/en/v1/base/math/#Base.mod.
 ↑ "Mathematics · The Julia Language". https://docs.julialang.org/en/v1/base/math/#Base.rem.
 ↑ "rem  Kotlin Programming Language" (in en). https://kotlinlang.org/api/latest/jvm/stdlib/kotlin/int/rem.html.
 ↑ "mod  Kotlin Programming Language" (in en). https://kotlinlang.org/api/latest/jvm/stdlib/kotlin/mod.html.
 ↑ "Chapter 3: The NASM Language". https://nasm.us/doc/nasmdoc3.html#section3.5.11.
 ↑ "OCaml library : Stdlib". https://ocaml.org/releases/4.13/api/Stdlib.html#VAL(mod).
 ↑ "OCaml library : Stdlib". https://ocaml.org/releases/4.13/api/Stdlib.html#VALmod_float.
 ↑ Perl documentation
 ↑ "PHP: Arithmetic Operators  Manual". https://www.php.net/manual/en/language.operators.arithmetic.php.
 ↑ "PHP: fmod  Manual". https://www.php.net/manual/en/function.fmod.php.
 ↑ "EuclideanRing". https://pursuit.purescript.org/packages/purescriptprelude/5.0.1/docs/Data.EuclideanRing#t:EuclideanRing.
 ↑ QuantumWriter. "Expressions" (in enus). https://docs.microsoft.com/enus/quantum/quantumqrexpressions?view=qsharppreview#numericexpressions.
 ↑ "F32  Rust". https://doc.rustlang.org/std/primitive.f32.html#method.rem_euclid.
 ↑ ^{43.0} ^{43.1} r6rs.org
 ↑ "Shell Command Language". https://pubs.opengroup.org/onlinepubs/9699919799/utilities/V3_chap02.html#tag_18_06_04.
 ↑ "Apple Developer Documentation". https://developer.apple.com/documentation/swift/binaryinteger/2885003.
 ↑ "Apple Developer Documentation". https://developer.apple.com/documentation/swift/floatingpoint/3017981remainder.
 ↑ "Apple Developer Documentation". https://developer.apple.com/documentation/swift/floatingpoint/3017995truncatingremainder.
 ↑ "Numerics — WebAssembly 1.1 (Draft 20220302)". https://webassembly.github.io/spec/core/exec/numerics.html#opiremu.
 ↑ "Numerics — WebAssembly 1.1 (Draft 20220302)". https://webassembly.github.io/spec/core/exec/numerics.html#opirems.
 ↑ ^{50.0} ^{50.1} "Mod". Wolfram Research. 2020. https://reference.wolfram.com/language/ref/Mod.html.
External links
 Modulorama, animation of a cyclic representation of multiplication tables (explanation in French)
de:Division mit Rest#Modulo