Modulo operation
In computing, the modulo operation finds the remainder after division of one number by another (sometimes called modulus).
Given two positive numbers, a (the dividend) and n (the divisor), a modulo n (abbreviated as a mod n) is the remainder of the Euclidean division of a by n. For instance, the expression "5 mod 2" would evaluate to 1 because 5 divided by 2 leaves 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 leaves a remainder of 0; there is nothing to subtract from 9 after multiplying 3 times 3. (Note that doing the division with a calculator will not show the result referred to here by this operation; the quotient will be expressed as a decimal fraction.)
Although typically performed with a and n both being integers, many computing systems allow other types of numeric operands. The range of numbers for an integer modulo of n is 0 to n − 1. (n mod 1 is always 0; n mod 0 is undefined, possibly resulting in a division by zero error in computer programming languages.) See modular arithmetic for an older and related convention applied in number theory.
When either a or n is negative, the naive definition breaks down and programming languages differ in how these values are defined.
Remainder calculation for the modulo operation
Language  Operator  Result has the same sign as 

ABAP  MOD  Always positive 
ActionScript  %  Dividend 
Ada  mod  Divisor 
rem  Dividend  
ALGOL68  ÷×, mod  Always positive 
AMPL  mod  Dividend 
APL   ^{}  Divisor 
AppleScript  mod  Dividend 
AWK  %  Dividend 
BASIC  Mod  Not defined 
bash  %  Dividend 
bc  %  Dividend 
C (ISO 1990)  %  Implementationdefined 
div  Dividend  
C++ (ISO 1998)  %  Implementationdefined^{[1]} 
div  Dividend  
C (ISO 1999)  % , div  Dividend^{[2]} 
C++ (ISO 2011)  % , div  Dividend 
C#  %  Dividend 
CLARION  %  Dividend 
Clojure  mod  Divisor 
COBOL^{}  FUNCTION MOD  Divisor 
CoffeeScript  %  Dividend 
%%  Divisor^{[3]}  
ColdFusion  %, MOD  Dividend 
Common Lisp  mod  Divisor 
rem  Dividend  
D  %  Dividend^{[4]} 
Dart  %  Always positive 
remainder()  Dividend  
Eiffel  \\  Dividend 
Erlang  rem  Dividend 
Euphoria  mod  Divisor 
remainder  Dividend  
F#  %  Dividend 
FileMaker  Mod  Divisor 
Forth  mod  implementation defined 
Fortran  mod  Dividend 
modulo  Divisor  
Frink  mod  Divisor 
GML (Game Maker)  mod  Dividend 
GDScript  %  Dividend 
Go  %  Dividend 
Haskell  mod  Divisor 
rem  Dividend  
Haxe  %  Dividend 
J   ^{}  Divisor 
Java  %  Dividend 
Math.floorMod  Divisor  
JavaScript  %  Dividend 
Julia  mod  Divisor 
rem  Dividend  
LibreOffice  =MOD()  Divisor 
Lua 5  %  Divisor 
Lua 4  mod(x,y)  Divisor 
Liberty BASIC  MOD  Dividend 
MathCad  mod(x,y)  Divisor 
Maple  e mod m  Always positive 
Mathematica  Mod  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^{} 
MUMPS  #  Divisor 
NASM NASMX  %  Unsigned Modulo Operator 
%%  Signed Modulo Operator  
Oberon  MOD  Divisor^{} 
OCaml  mod  Dividend 
Occam  \  Dividend 
Pascal (Delphi)  mod  Dividend 
Pascal (ISO7185 and ISO10206)  mod  Always positive 
Perl  %  Divisor^{} 
PHP  %  Dividend 
PIC Basic Pro  \\  Dividend 
PL/I  mod  Divisor (ANSI PL/I) 
PowerShell  %  Dividend 
Progress  modulo  Dividend 
Prolog (ISO 1995)  mod  Divisor 
rem  Dividend  
Python  %  Divisor 
math.fmod  Dividend  
Racket  remainder  Dividend 
RealBasic  MOD  Dividend 
R  %%  Divisor 
REXX  //  Dividend 
RPG  %REM  Dividend 
Ruby  %, modulo()  Divisor 
remainder()  Dividend  
Rust  %  Dividend 
Scala  %  Dividend 
Scheme  modulo  Divisor 
remainder  Dividend  
Scheme R^{6}RS  mod  Always positive^{[5]} 
mod0  Closest to zero^{[5]}  
Seed7  mod  Divisor 
rem  Dividend  
SenseTalk  modulo  Divisor 
rem  Dividend  
Smalltalk  \\  Divisor 
rem:  Dividend  
SPIN  //  Divisor 
SQL (SQL:1999)  mod(x,y)  Dividend 
SQL (SQL:2012)  %  Dividend 
Standard ML  mod  Divisor 
Int.rem  Dividend  
Stata  mod(x,y)  Always positive 
Swift  %  Dividend 
Tcl  %  Divisor 
Torque Game Engine  %  Dividend 
Turing  mod  Divisor 
Verilog (2001)  %  Dividend 
VHDL  mod  Divisor 
rem  Dividend  
Visual Basic  Mod  Dividend 
x86 Assembly  IDIV  Dividend 
Xbase++  %  Dividend 
Mod()  Divisor  
Z3 theorem prover  div , mod  Always positive 
Language  Operator  Result has the same sign as 

ABAP  MOD  Always positive 
C (ISO 1990)  fmod  Dividend^{[6]} 
C (ISO 1999)  fmod  Dividend 
remainder  Closest to zero  
C++ (ISO 1998)  std::fmod  Dividend 
C++ (ISO 2011)  std::fmod  Dividend 
std::remainder  Closest to zero  
C#  %  Dividend 
Common Lisp  mod  Divisor 
rem  Dividend  
D  %  Dividend 
Dart  %  Always positive 
remainder()  Dividend  
F#  %  Dividend 
Fortran  mod  Dividend 
modulo  Divisor  
Go  math.Mod  Dividend 
Haskell (GHC)  Data.Fixed.mod'  Divisor 
Java  %  Dividend 
JavaScript  %  Dividend 
Microsoft Excel  =MOD()  Divisor 
OCaml  mod_float  Dividend 
Perl  POSIX::fmod  Dividend 
Perl6  %  Divisor 
PHP  fmod  Dividend 
Python  %  Divisor 
math.fmod  Dividend  
REXX  //  Dividend 
Ruby  %, modulo()  Divisor 
remainder()  Dividend  
Scheme R^{6}RS  flmod  Always positive 
flmod0  Closest to zero  
Standard ML  Real.rem  Dividend 
Swift  %  Dividend 
Xbase++  %  Dividend 
Mod()  Divisor 
In mathematics the result of the modulo operation is 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 and/or the underlying hardware.
In nearly all computing systems, the quotient q and the remainder r of a divided by n satisfy

(1)
However, this still leaves a sign ambiguity if the remainder is nonzero: there are two possible choices for the remainder, one negative and the other positive, and there are also two possible choices for the quotient. Usually, in number theory, the positive remainder is always chosen, but programming languages choose depending on the language and the signs of a and/or n.^{} Standard Pascal and Algol68 give a positive remainder (or 0) even for negative divisors, and some programming languages, such as C90, leave it up to the implementation when either of n or a is negative. See the table for details. a modulo 0 is undefined in the majority of systems, although some do define it to be a.
 Many implementations use truncated division, where the quotient is defined by 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^{[7]} 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^{[8]} describes the Euclidean definition in which the remainder is always nonnegative, 0 ≤ r, and is therefore consistent with the Euclidean division algorithm. This convention is denoted Always positive in the table. 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. Therefore, the sign of the remainder is chosen so as to be closest 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^{[9]}
Common pitfalls
When the result of a modulo operation has the sign of the dividend, it can sometimes lead to surprising mistakes:
For example, to test whether an integer is odd, one might be inclined to test whether the remainder by 2 is equal to 1:
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 it is not 0 (because remainder 0 is the same regardless of the signs):
bool is_odd(int n) {
return n % 2 != 0;
}
Or, by understanding in the first place that for any odd number, the modulo remainder may be either 1 or −1:
bool is_odd(int n) {
return n % 2 == 1  n % 2 == 1;
}
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
or equivalent, for environments lacking a mod() function (note that 'int' inherently produces the truncated value of a/n)
a  (n * int(a/n))
.
Performance issues
Modulo operations might be implemented such that a division with a remainder is calculated each time. For special cases, there are faster alternatives on some hardware. 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.^{[10]}
Optimizing compilers may recognize expressions of the form expression % constant
where constant
is a power of two and automatically implement them as expression & (constant1)
. This can allow the programmer to write clearer code without compromising performance. This optimization is not possible for the languages whose modulo have 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.
Equivalencies
Some modulo operations can be factored or expanded similar 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. Not defined otherwise.
 Inverse multiplication: [(ab mod n)(b^{−1} mod n)] mod n = a mod n.
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.
 Modular exponentiation
Notes
 ^ Perl usually uses arithmetic modulo operator that is machineindependent. See the Perl documentation for exceptions and examples.
 ^ 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 not defined.
 ^ As implemented in ACUCOBOL, Micro Focus COBOL, and possibly others.
 ^ ^ Argument order is flipped, i.e.
αω
computes , the reminder when dividingω
byα
.
References
 ↑ "ISO/IEC 14882:2003 : Programming languages  C++". 5.6.4: ISO, IEC. 2003.. "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".
 ↑ openstd.org, section 6.5.5
 ↑ CoffeeScript operators
 ↑ "Expressions". D Programming Language 2.0. Digital Mars. Retrieved 29 July 2010.
 1 2 r6rs.org
 ↑ "ISO/IEC 9899:1990 : Programming languages  C". 7.5.6.4: ISO, IEC. 1990.. "The
fmod
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
.".  ↑ 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.
 ↑ 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".