# Modulo operation

Quotient (red) and remainder (green) functions using different algorithms

In computing, the modulo operation finds the remainder of 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

Integer modulo operators in various programming languages
LanguageOperatorResult has the same sign as
ActionScript%Dividend
remDividend
ASPModNot defined
ALGOL-68÷×, modAlways positive
AMPLmodDividend
APL|Divisor
AppleScriptmodDividend
AWK%Dividend
BASICModNot defined
bash%Dividend
bc%Dividend
C (ISO 1990)%Implementation defined
C++ (ISO 1998)%Implementation defined[1]
C (ISO 1999)%Dividend[2]
C++ (ISO 2011)%Dividend
C#%Dividend
CLARION%Dividend
ClojuremodDivisor
COBOL[1]FUNCTION MODDivisor
ColdFusion%, MODDividend
Common LispmodDivisor
remDividend
D%Dividend[3]
Dart%Always Positive
remainder()Dividend
Eiffel\\Dividend
ErlangremDividend
EuphoriamodDivisor
remainderDividend
F#%Dividend
FileMakerModDivisor
Forthmodimplementation defined
FortranmodDividend
moduloDivisor
FrinkmodDivisor
GML (Game Maker)modDividend
GDScriptmodDividend
Go%Dividend
remDividend
Haxe%Dividend
J|~Divisor
Java%Dividend
Math.floorModDivisor
JavaScript%Dividend
JuliamodDivisor
remDividend
LibreOffice=MOD()Divisor
Lua 5%Divisor
Lua 4mod(x,y)Divisor
Liberty BASICMODDividend
Maplee mod mAlways positive
MathematicaModDivisor
MATLABmodDivisor
remDividend
MaximamodDivisor
remainderDividend
Maya Embedded Language%Dividend
Microsoft Excel=MOD()Divisor
MinitabMODDivisor
mksh%Dividend
Modula-2MODDivisor[2]
MUMPS#Divisor
NASM NASMX%Unsigned Modulo Operator
%%Signed Modulo Operator
OberonMODDivisor[3]
OCamlmodDividend
Occam\Dividend
Pascal (Delphi)modDividend
Pascal (ISO-7185 and ISO-10206)modAlways positive
Perl%Divisor[4]
PHP%Dividend
PIC Basic Pro\\Dividend
PL/ImodDivisor (ANSI PL/I)
PowerBuildermod(x,y) ?
PowerShell%Dividend
ProgressmoduloDividend
Prolog (ISO 1995)modDivisor
remDividend
Python%Divisor
math.fmodDividend
RacketremainderDividend
RealBasicMODDividend
R%%Divisor
REXX//Dividend
RPG%REMDividend
Ruby%, modulo()Divisor
remainder()Dividend
Scala%Dividend
SchememoduloDivisor
remainderDividend
Scheme R6RSmodAlways positive[4]
mod0Closest to zero[4]
Seed7modDivisor
remDividend
SenseTalkmoduloDivisor
remDividend
Smalltalk\\Divisor
rem:Dividend
SQL (SQL:1999)mod(x,y)Dividend
Standard MLmodDivisor
Int.remDividend
Statamod(x,y)Always positive
Swift%Dividend
Tcl%Divisor
Torque Game Engine%Dividend
TuringmodDivisor
Verilog (2001)%Dividend
VHDLmodDivisor
remDividend
Visual BasicModDividend
x86 AssemblyIDIVDividend
Xbase++%Dividend
Mod()Divisor
Floating-point modulo operators in various programming languages
LanguageOperatorResult has the same sign as
C (ISO 1990)fmod ?
C (ISO 1999)fmodDividend
remainderClosest to zero
C++ (ISO 1998)std::fmod ?
C++ (ISO 2011)std::fmodDividend
std::remainderClosest to zero
C#%Dividend
Common LispmodDivisor
remDividend
D%Dividend
Dart%Always Positive
remainder()Dividend
F#%Dividend
FortranmodDividend
moduloDivisor
Gomath.ModDividend
Java%Dividend
JavaScript%Dividend
Microsoft Excel=MOD()Divisor
OCamlmod_floatDividend
PerlPOSIX::fmodDividend
Perl6%Divisor
PHPfmodDividend
Python%Divisor
math.fmodDividend
REXX//Dividend
Ruby%, modulo()Divisor
remainder()Dividend
Scheme R6RSflmodAlways positive
flmod0Closest to zero
Standard MLReal.remDividend
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 satisfy

$q \in \mathbb{Z}$
$a = n \times q + r\,$
$\left| r \right| < \left| n \right|.$

This means that, 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 n.[5] Pascal and Algol68 give a positive remainder (or 0) even for negative divisors, and some programming languages, such as C89, do not define a result if 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), in other words it is the first integer in the direction of 0 from the exact rational quotient, and the remainder by r=an q. Informally speaking the quotient is "rounded towards zero", and the remainder therefore has the same sign as the dividend.

Knuth[5] described floored division where the quotient is defined by the floor function q=floor(a/n) and the remainder r is

$r = a - nq = a - n \left\lfloor {a \over n} \right\rfloor.$

Here the quotient is always rounded downwards (even if it is already negative) and the remainder has the same sign as the divisor.

Raymond T. Boute[6] introduces the Euclidean definition, which is the one in which the remainder is always positive or 0, and is therefore consistent with the division algorithm (see Euclidean division). This definition is marked as "Always positive" in the table. Let q be the integer quotient of a and n, then:

$q \in \mathbb{Z}$
$a = n \times q + r\,$
$0 \leq r < |n|.$

Two corollaries are that

$n > 0 \Rightarrow q = \left\lfloor \frac{a}{n} \right\rfloor$
$n < 0 \Rightarrow q = \left\lceil \frac{a}{n} \right\rceil,$

or, equivalently,

$q = \sgn(n) \left\lfloor \frac{a}{\left|n\right|} \right\rfloor.$

As described by Leijen,[7]

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.

Common Lisp also defines round- and ceiling-division where the quotient is given by q=round(a/n), q=ceil(a/n). IEEE 754 defines a remainder function where the quotient is a/n rounded according to the round to nearest convention.

## 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 % 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; }

## Modulo operation expression

Some calculators have a mod() function button, and many programming languages have a mod() function or similar, 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 floor 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 % 2n == x & (2n - 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. [8]

Optimizing compilers may recognize expressions of the form expression % constant where constant is a power of two and automatically implement them as expression & (constant-1). This can allow the programmer to write clearer code without compromising performance. (Note: This will not work for the languages whose modulo have the sign of the dividend (including C), because if the dividend is negative, the modulo will be negative; however, expression & (constant-1) will always produce a positive result. So special treatment has to be made when the dividend can be negative.)

## 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:
• $(a\,\bmod\,n)\,\bmod\,n = a\,\bmod\,n$
• $n^x\,\bmod\,n = 0$ for all positive integer values of $x$.
• If $n$ is a prime number which is not a divisor of $b$, then $ab^{n-1}\,\bmod\,n = a\,\bmod\,n$, due to Fermat's little theorem.
• Inverse:
• $((-a\,\bmod\,n) + (a\,\bmod\,n))\,\bmod\,n =0$
• $b^{-1}\,\bmod\,n$ 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: $((b^{-1}\,\bmod\,n) \, (b\,\bmod\,n))\,\bmod\,n =1$.
• Distributive:
• $(a+b)\,\bmod\,n = ((a\,\bmod\,n)+(b\,\bmod\,n))\,\bmod\,n$
• $ab\,\bmod\,n = ((a\,\bmod\,n)\,(b\,\bmod\,n))\,\bmod\,n$
• Division (definition): $\frac{a}{b}\,\bmod\,n = ((a\,\bmod\,n)(b^{-1}\,\bmod\,n))\,\bmod\,n$, when the right hand side is defined. Not defined otherwise.
• Inverse Multiplication: $((ab\,\bmod\,n)\,(b^{-1}\,\bmod\,n))\,\bmod\,n = a\,\bmod\,n$

## Notes

• ^ Perl usually uses arithmetic modulo operator that is machine-independent. 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.

## References

1. ^ "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 implementation-defined".
2. ^ open-std.org, section 6.5.5
3. ^ "Expressions". D Programming Language 2.0. Digital Mars. Retrieved 29 July 2010.
4. ^ a b r6rs.org
5. ^ Knuth, Donald. E. (1972). The Art of Computer Programming. Addison-Wesley.
6. ^ Boute, Raymond T. (April 1992). "The Euclidean definition of the functions div and mod". ACM Transactions on Programming Languages and Systems (TOPLAS) (ACM Press (New York, NY, USA)) 14 (2): 127–144. doi:10.1145/128861.128862.
7. ^ Leijen, Daan (December 3, 2001). "Division and Modulus for Computer Scientists" (PDF). Retrieved 2006-08-27.
8. ^ Horvath, Adam (July 5, 2012). "Faster division and modulo operation - the power of two".