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 (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.^{[1]} The modulo operation is to be distinguished from the symbol mod, which refers to the modulus^{[2]} (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.
(Here, notice that doing division with a calculator will not show the result of the modulo operation, and that the quotient will be expressed as a decimal fraction if a nonzero remainder is involved.)
Although typically performed with a and n both being integers, many computing systems now allow other types of numeric operands. The range of numbers for an integer modulo 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 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 nonnegative integer that belongs to that class (i.e., the remainder of the Euclidean division).^{[3]} 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:
(1)
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.^{[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 q = trunc(a/n) and thus according to equation (1) the remainder would have 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.
 Donald Knuth^{[4]} described floored division where the quotient is defined by the floor function q = ⌊a/n⌋ 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.
 Raymond T. Boute^{[5]} 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,
or equivalently
where sgn is the sign function, and thus
 Common Lisp also defines rounddivision and ceilingdivision where the quotient is given by q = round(a/n) and q = ⌈a/n⌉ respectively.
 IEEE 754 defines a remainder function where the quotient is a/n rounded according to the round to nearest convention. Thus, the sign of the remainder is chosen to be nearest to zero.
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^{[6]}
However, Boute concentrates on the properties of the modulo operation itself, and does not rate the fact that truncated division shows the symmetry (a) div n = (a div n) and a div (n) = (a div n), which is similar to ordinary division. As neither floor division nor Euclidean division offer this symmetry, Boute's judgement is at least incomplete.^{[citation needed]}^{[original research?]}
Common pitfallsEdit
When the result of a modulo operation has the sign of the dividend, 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;
}
NotationEdit
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
or equivalent, for environments lacking a mod() function ('int' inherently produces the truncated value of a/n)
a  (n * int(a/n))
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:
x % 2^{n} == x & (2^{n}  1)
Examples (assuming x is a positive integer):
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]}
Optimizing compilers 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.
Properties (identities)Edit
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.
 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). Undefined otherwise.
 Inverse multiplication: [(ab mod n)(b^{−1} mod n)] mod n = a mod n.
In programming languagesEdit
Language  Operator  Result has same sign as 

ABAP  MOD

Nonnegative always 
ActionScript  %

Dividend 
Ada  mod

Divisor 
rem

Dividend  
ALGOL 68  ÷× , mod

Nonnegative always 
AMPL  mod

Dividend 
APL   ^{[2]} 
Divisor 
AppleScript  mod

Dividend 
AutoLISP  (rem d n)

Dividend 
AWK  %

Dividend 
BASIC  Mod

Undefined 
bash  %

Dividend 
bc  %

Dividend 
C (ISO 1990)  %

Implementationdefined 
div

Dividend  
C++ (ISO 1998)  %

Implementationdefined^{[8]} 
div

Dividend  
C (ISO 1999)  % , div

Dividend^{[9]} 
C++ (ISO 2011)  % , div

Dividend 
C#  %

Dividend 
Clarion  %

Dividend 
Clean  rem

Dividend 
Clojure  mod

Divisor 
rem

Dividend  
COBOL^{[3]}  FUNCTION MOD

Divisor 
CoffeeScript  %

Dividend 
%%

Divisor^{[10]}  
ColdFusion  % , MOD

Dividend 
Common Lisp  mod

Divisor 
rem

Dividend  
Crystal  %

Dividend 
D  %

Dividend^{[11]} 
Dart  %

Nonnegative always 
remainder()

Dividend  
Eiffel  \\

Dividend 
Elixir  rem

Dividend 
Elm  modBy

Divisor 
remainderBy

Dividend  
Erlang  rem

Dividend 
Euphoria  mod

Divisor 
remainder

Dividend  
F#  %

Dividend 
Factor  mod

Dividend 
FileMaker  Mod

Divisor 
Forth  mod

implementation defined 
fm/mod

Divisor  
sm/rem

Dividend  
Fortran  mod

Dividend 
modulo

Divisor  
Frink  mod

Divisor 
GameMaker Studio (GML)  mod , %

Dividend 
GDScript  %

Dividend 
Go  %

Dividend 
Groovy  %

Dividend 
Haskell  mod

Divisor 
rem

Dividend  
Haxe  %

Dividend 
J   ^{[4]} 
Divisor 
Java  %

Dividend 
Math.floorMod

Divisor  
JavaScript  %

Dividend 
Julia  mod

Divisor 
% , rem

Dividend  
Kotlin  % , rem

Dividend 
ksh  %

Dividend 
LabVIEW  mod

Dividend 
LibreOffice  =MOD()

Divisor 
Logo  MODULO

Divisor 
REMAINDER

Dividend  
Lua 5  %

Divisor 
Lua 4  mod(x,y)

Divisor 
Liberty BASIC  MOD

Dividend 
Mathcad  mod(x,y)

Divisor 
Maple  e mod m

Nonnegative always 
Mathematica  Mod[a, b]

Divisor 
MATLAB  mod

Divisor 
rem

Dividend  
Maxima  mod

Divisor 
remainder

Dividend  
Maya Embedded Language  %

Dividend 
Microsoft Excel  =MOD()

Divisor 
Minitab  MOD

Divisor 
mksh  %

Dividend 
Modula2  MOD

Divisor 
REM

Dividend  
MUMPS  #

Divisor 
Netwide Assembler (NASM, NASMX)  % , div

Modulo operator unsigned 
%%

Modulo operator signed  
Nim  mod

Dividend 
Oberon  MOD

Divisor^{[5]} 
ObjectiveC  %

Dividend 
Object Pascal, Delphi  mod

Dividend 
OCaml  mod

Dividend 
Occam  \

Dividend 
Pascal (ISO7185 and 10206)  mod

Nonnegative always^{[6]} 
Programming Code Advanced (PCA)  \

Undefined 
Perl  %

Divisor^{[7]} 
Phix  mod

Divisor 
remainder

Dividend  
PHP  %

Dividend 
PIC BASIC Pro  \\

Dividend 
PL/I  mod

Divisor (ANSI PL/I) 
PowerShell  %

Dividend 
Programming Code (PRC)  MATH.OP  'MOD; (\)'

Undefined 
Progress  modulo

Dividend 
Prolog (I SO 1995)  mod

Divisor 
rem

Dividend  
PureBasic  % , Mod(x,y)

Dividend 
PureScript  `mod`

Divisor 
Python  %

Divisor 
Q#  %

Dividend^{[12]} 
R  %%

Divisor 
RealBasic  MOD

Dividend 
Reason  mod

Dividend 
Racket  modulo

Divisor 
remainder

Dividend  
Rexx  //

Dividend 
RPG  %REM

Dividend 
Ruby  % , modulo()

Divisor 
remainder()

Dividend  
Rust  %

Dividend 
rem_euclid()

Divisor  
SAS  MOD

Dividend 
Scala  %

Dividend 
Scheme  modulo

Divisor 
remainder

Dividend  
Scheme R^{6}RS  mod

Nonnegative always^{[13]} 
mod0

Nearest to zero^{[13]}  
Scratch  mod

Divisor 
Seed7  mod

Divisor 
rem

Dividend  
SenseTalk  modulo

Divisor 
rem

Dividend  
Shell  %

Dividend 
Smalltalk  \\

Divisor 
rem:

Dividend  
Snap!  mod

Divisor 
Spin  //

Divisor 
Solidity  %

Divisor 
SQL (SQL:1999)  mod(x,y)

Dividend 
SQL (SQL:2011)  %

Dividend 
Standard ML  mod

Divisor 
Int.rem

Dividend  
Stata  mod(x,y)

Nonnegative always 
Swift  %

Dividend 
Tcl  %

Divisor 
TypeScript  %

Dividend 
Torque  %

Dividend 
Turing  mod

Divisor 
Verilog (2001)  %

Dividend 
VHDL  mod

Divisor 
rem

Dividend  
VimL  %

Dividend 
Visual Basic  Mod

Dividend 
WebAssembly  i32.rem_s , i64.rem_s

Dividend 
x86 assembly  IDIV

Dividend 
XBase++  %

Dividend 
Mod()

Divisor  
Z3 theorem prover  div , mod

Nonnegative always 
Language  Operator  Result has same sign as 

ABAP  MOD

Nonnegative always 
C (ISO 1990)  fmod

Dividend^{[14]} 
C (ISO 1999)  fmod

Dividend 
remainder

Nearest to zero  
C++ (ISO 1998)  std::fmod

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

Dividend 
std::remainder

Nearest to zero  
C#  %

Dividend 
Common Lisp  mod

Divisor 
rem

Dividend  
D  %

Dividend 
Dart  %

Nonnegative always 
remainder()

Dividend  
F#  %

Dividend 
Fortran  mod

Dividend 
modulo

Divisor  
Go  math.Mod

Dividend 
Haskell (GHC)  Data.Fixed.mod'

Divisor 
Java  %

Dividend 
JavaScript  %

Dividend 
ksh  fmod

Dividend 
LabVIEW  mod

Dividend 
Microsoft Excel  =MOD()

Divisor 
OCaml  mod_float

Dividend 
Perl  POSIX::fmod

Dividend 
Raku  %

Divisor 
PHP  fmod

Dividend 
Python  %

Divisor 
math.fmod

Dividend  
Rexx  //

Dividend 
Ruby  % , modulo()

Divisor 
remainder()

Dividend  
Rust  %

Dividend 
rem_euclid()

Divisor  
Scheme R^{6}RS  flmod

Nonnegative always 
flmod0

Nearest to zero  
Scratch  mod

Dividend 
Standard ML  Real.rem

Dividend 
Swift  truncatingRemainder(dividingBy:)

Dividend 
XBase++  %

Dividend 
Mod()

Divisor 
GeneralizationsEdit
Modulo with offsetEdit
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:^{[15]} 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:
 a mod_{d} n = .
(This is easy to see. Let . We first show that x mod n = a mod n. It is in genereal true that (a+bn) mod n = a mod n for all integers b; thus, this is true also in the particular case when b = ; but that means that , 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 ≤ n1 (see Euclidean division). Then , thus . 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^{[15]} Mod[a, n, d]
.
See alsoEdit
 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)
NotesEdit
 ^ Perl usually uses arithmetic modulo operator that is machineindependent. For examples and exceptions, see the Perl documentation on multiplicative operators.^{[16]}
 ^ Mathematically, these two choices are but two of the infinite number of choices available for the inequality satisfied by a remainder.
 ^ Divisor must be positive, otherwise undefined.
 ^ As implemented in ACUCOBOL, Micro Focus COBOL, and possible others.
 ^ ^ Argument order reverses, i.e.,
αω
computes , the remainder when dividingω
byα
.  ^ As discussed by Boute, ISO Pascal's definitions of
div
andmod
do not obey the Division Identity, and are thus fundamentally broken.
ReferencesEdit
 ^ "The Definitive Glossary of Higher Mathematical Jargon: Modulo". Math Vault. 20190801. Retrieved 20200827.
 ^ Weisstein, Eric W. "Congruence". mathworld.wolfram.com. Retrieved 20200827.
 ^ Caldwell, Chris. "residue". Prime Glossary. Retrieved August 27, 2020.
 ^ Knuth, Donald. E. (1972). The Art of Computer Programming. AddisonWesley.
 ^ 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. hdl:1854/LU314490.
 ^ Leijen, Daan (December 3, 2001). "Division and Modulus for Computer Scientists" (PDF). Retrieved 20141225.
 ^ Horvath, Adam (July 5, 2012). "Faster division and modulo operation  the power of two".
 ^ "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
Cite journal requiresjournal=
(help)  ^ "C99 specification (ISO/IEC 9899:TC2)" (PDF). 20050506. sec. 6.5.5 Multiplicative operators. Retrieved 16 August 2018.
 ^ CoffeeScript operators
 ^ "Expressions". D Programming Language 2.0. Digital Mars. Retrieved 29 July 2010.
 ^ QuantumWriter. "Expressions". docs.microsoft.com. Retrieved 20180711.
 ^ ^{a} ^{b} r6rs.org
 ^ "ISO/IEC 9899:1990: Programming languages – C". ISO, IEC. 1990. sec. 7.5.6.4.
The
Cite journal requiresfmod
function returns the valuex  i * y
, for some integeri
such that, ify
is nonzero, the result as the same sign asx
and magnitude less than the magnitude ofy
.journal=
(help)  ^ ^{a} ^{b} "Mod". Wolfram Language & System Documentation Center. Wolfram Research. 2020. Retrieved April 8, 2020.
 ^ Perl documentation
External linksEdit
 Modulorama, animation of a cyclic representation of multiplication tables (explanation in French)