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# Modulo operation

In computing, the modulo operation finds the remainder of division of one number by another.

Given two numbers, (the dividend) and (the divisor), a modulo n (abbreviated as a mod n) is the remainder, on division of a by n. For instance, the expression "7 mod 3" would evaluate to 1, while "9 mod 3" would evaluate to 0. Although typically performed with a and n both being integers, many computing systems allow other types of numeric operands.

See modular arithmetic for an older and related convention applied in number theory.

## Remainder calculation for the modulo operation

Modulo operators in various programming languages
Language Operator Result has the same sign as
ActionScript % Dividend
rem Dividend
ASP Mod Not defined
C (ISO 1990) % Not defined
C (ISO 1999) % Dividend
C++ % Not defined
C# % Dividend
ColdFusion MOD Dividend
Common Lisp mod Divisor
rem Dividend
Eiffel Dividend
Microsoft Excel =MOD() Divisor
Euphoria remainder Dividend
FileMaker Mod Divisor
Fortran mod Dividend
modulo Divisor
GML (Game Maker) mod Dividend
rem Dividend
J |~ Divisor
Java % Dividend
JavaScript % Dividend
Lua % Divisor
Mathematica Mod Divisor
MATLAB mod Divisor
rem Dividend
MySQL MOD
%
Dividend
Objective Caml mod Not defined
Occam Dividend
Pascal (Delphi) mod Dividend
Perl % Divisor
PHP % Dividend
PL/I mod Divisor (ANSI PL/I)
Prolog (ISO 1995) mod Divisor
rem Dividend
Python % Divisor
QBasic MOD Dividend
R %% Divisor
RPG %REM Dividend
Ruby % Divisor
Scheme modulo Divisor
SenseTalk modulo Divisor
rem Dividend
Smalltalk Divisor
Tcl % Divisor
Verilog (2001) % Dividend
VHDL mod Divisor
rem Dividend
Visual Basic Mod Dividend
There are various ways of defining a remainder, and computers and calculators have various ways of storing and representing numbers, so what exactly constitutes the result of a modulo operation depends on the programming language and/or the underlying hardware.

In nearly all computing systems, the quotient and the remainder satisfy

• $a = q n + r,$
• $q,$ is an integer
• $0 leq |r| < |n|$

Pascal and Algol68 do not satisfy these for negative divisors. Some programming languages, such as C89, don't even 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) and the remainder by r=a-n q. With this definition the quotient is rounded towards zero and the remainder has the same sign as the dividend.

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

$r = a - n leftlfloor \left\{a over n\right\} rightrfloor.$
Here the quotient rounds towards negative infinity and the remainder has the same sign as the divisor.

Raymond T. Boute introduces the Euclidean definition which is consistent with the division algorithm. Let q be the integer quotient of a and n, then:

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

Two corollaries are that

$n > 0 to q = leftlfloor a div n rightrfloor$
$n < 0 to q = leftlceil a div n rightrceil.$

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.

Common Lisp also defines round- and ceiling-division where the quotient is given by , . IEEE 754 defines a remainder function where the quotient is rounded according to the round to nearest convention.

## 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`.

## Performance issues

Modulo operations might be implemented such that division with 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 an 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.

In the C programming language, compiling with heavy speed optimizations will typically (depending on compiler and hardware) automatically convert modulo operations to bitwise AND in the assembly file.

In some compilers, the modulo operation is implemented as `mod(a, n) = a - n * floor(a / n)`. When performing both modulo and division on the same numbers, one can get the same result somewhat more efficiently by avoiding the actual modulo operator, and using the formula above on the result, avoiding an additional division operation.

• Modulo — many uses of the word "modulo", all of which grew out of Carl F. Gauss's introduction of modular arithmetic in 1801.

## References

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