Modulo

In computing and mathematics, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, the latter being called the modulus of the operation.
Given two positive numbers and, modulo is the remainder of the Euclidean division of by, where is the dividend and is the divisor.
For example, the expression "5 mod 2" evaluates 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 9 divided by 3 has a quotient of 3 and a remainder of 0.
Although typically performed with and both being integers, many computing systems now allow other types of numeric operands. The range of values for an integer modulo operation of is 0 to. mod 1 is always 0.
When exactly one of or is negative, the basic 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 non-negative integer that belongs to that class. 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 and the remainder of divided by satisfy the following conditions:
This still leaves a sign ambiguity if the remainder is non-zero: two possible choices for the remainder occur, one negative and the other positive; that choice determines which of the two consecutive quotients must be used to satisfy equation. In number theory, the positive remainder is always chosen, but in computing, programming languages choose depending on the language and the signs of or. Standard Pascal and ALGOL 68, for example, give a positive remainder even for negative divisors, and some programming languages, such as C90, leave it to the implementation when either of or is negative. Some systems leave modulo 0 undefined, though others define it as.
If both the dividend and divisor are positive, then the truncated, floored, and Euclidean definitions agree.
If the dividend is positive and the divisor is negative, then the truncated and Euclidean definitions agree.
If the dividend is negative and the divisor is positive, then the floored and Euclidean definitions agree.
If both the dividend and divisor are negative, then the truncated and floored definitions agree.
However, truncated division satisfies the identity.

Notation

Some calculators have a function button, and many programming languages have a similar function, expressed as, for example. Some also support expressions that use "%", "mod", or "Mod" as a modulo or remainder operator, such as or.
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, 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

But in a language where modulo has the sign of the dividend, that is incorrect, because when is negative and odd, mod 2 returns −1, and the function returns false.
One correct alternative is to test that the remainder is not 0 :

bool is_odd

Or with the binary arithmetic:

bool is_odd

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 :
Examples:
In devices and software that implement bitwise operations more efficiently than modulo, these alternative forms can result in faster calculations.
Compiler optimizations may recognize expressions of the form where is a power of two and automatically implement them as, 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, unless the dividend is of an unsigned integer type. This is because, if the dividend is negative, the modulo will be negative, whereas will always be positive. For these languages, the equivalence x % 2ⁿ x < 0 ? x | ~ : x & has to be used instead, expressed using bitwise OR, NOT and AND operations.
Optimizations for general constant-modulus operations also exist by calculating the division first using the constant-divisor 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. The properties involving multiplication, division, and exponentiation generally require that and are integers.

Identity:
*.
* for all positive integer values of.
* If is a prime number which is not a divisor of, then, due to Fermat's little theorem.
Inverse:
*.
* denotes the modular multiplicative inverse, which is defined if and only if and are relatively prime, which is the case when the left hand side is defined:.
Distributive:
*.
*.
Division : , when the right hand side is defined, and undefined otherwise.
Inverse multiplication: .
In programming languages

Language	Operator	Integer	Floating-point	Definition
ABAP				Euclidean
ActionScript				Truncated
Ada				Floored
Ada				Truncated
ALGOL 68	,,,, `mod`			Euclidean
AMPL				Truncated
APL	`\|`			Floored
AppleScript				Truncated
AutoLISP				Truncated
AWK				Truncated
BASIC				Varies by implementation
bc				Truncated
CC++	,			Truncated
CC++				Truncated
CC++				Rounded
C#				Truncated
C#				Rounded
Clarion				Truncated
Clean				Truncated
Clojure				Floored
Clojure				Truncated
COBOL				Floored
COBOL				Truncated
CoffeeScript				Truncated
CoffeeScript				Floored
ColdFusion	,			Truncated
Common Intermediate Language				Truncated
Common Intermediate Language
Common Lisp				Floored
Common Lisp				Truncated
Crystal	,			Floored
Crystal				Truncated
CSS				Floored
CSS				Truncated
D				Truncated
Dart				Euclidean
Dart				Truncated
Eiffel				Truncated
Elixir				Truncated
Elixir				Floored
Elm				Floored
Elm				Truncated
Erlang				Truncated
Erlang				Truncated
Euphoria				Truncated
Euphoria				Floored
F#				Truncated
F#				Rounded
Factor				Truncated
Factor				Euclidean
FileMaker				Floored
Forth				Implementation defined
Forth				Floored
Forth				Truncated
Fortran				Truncated
Fortran				Floored
Frink				Floored
Full BASIC				Floored
Full BASIC				Truncated
GLSL				Undefined
GLSL				Floored
GameMaker Studio	,			Truncated
GDScript				Truncated
GDScript				Euclidean
GDScript				Truncated
GDScript				Euclidean
Go				Truncated
Go				Truncated
Go				Euclidean
Go				Truncated
Groovy				Truncated
Haskell				Floored
Haskell				Truncated
Haskell				Floored
Haxe				Truncated
HLSL				Undefined
J	`\|`			Floored
Java				Truncated
Java				Floored
JavaScriptTypeScript				Truncated
Julia				Floored
Julia	,			Truncated
Kotlin	,			Truncated
Kotlin				Floored
ksh				Truncated
ksh				Truncated
LabVIEW				Truncated
LibreOffice				Floored
Logo				Floored
Logo				Truncated
Lua 5				Floored
Lua 4				Truncated
Liberty BASIC				Truncated
Mathcad				Floored
Maple	,			Euclidean
Maple				Rounded
Maple				Rounded
Mathematica				Floored
MATLAB				Floored
MATLAB				Truncated
Maxima				Floored
Maxima				Truncated
Maya Embedded Language				Truncated
Microsoft Excel				Floored
Minitab				Floored
Modula-2				Floored
Modula-2				Truncated
MUMPS				Floored
Netwide Assembler	,
Netwide Assembler				Implementation-defined
Nim				Truncated
Oberon				Floored-like
Objective-C				Truncated
Object Pascal, Delphi				Truncated
OCaml				Truncated
OCaml				Truncated
Occam				Truncated
Pascal				Euclidean-like
Perl				Floored
Perl				Truncated
PHP				Truncated
PHP				Truncated
PIC BASIC Pro				Truncated
PL/I				Floored
PowerShell				Truncated
Programming Code				Undefined
Progress				Truncated
Prolog				Floored
Prolog				Truncated
PureBasic	,			Truncated
PureScript				Euclidean
Pure Data				Truncated
Pure Data				Floored
Python				Floored
Python				Truncated
Python				Rounded
Q#				Truncated
R				Floored
Racket				Floored
Racket				Truncated
Raku				Floored
RealBasic				Truncated
Reason				Truncated
Rexx				Truncated
RPG				Truncated
Ruby	,			Floored
Ruby				Truncated
Rust				Truncated
Rust				Euclidean
SAS				Truncated
Scala				Truncated
Scheme				Floored
Scheme				Truncated
Scheme R⁶RS				Euclidean
Scheme R⁶RS				Rounded
Scheme R⁶RS				Euclidean
Scheme R⁶RS				Rounded
Scratch				Floored
Seed7				Floored
Seed7				Truncated
SenseTalk				Floored
SenseTalk				Truncated
				Truncated
Smalltalk				Floored
Smalltalk				Truncated
Snap!				Floored
Spin				Floored
Solidity				Truncated
SQL				Truncated
SQL				Truncated
Standard ML				Floored
Standard ML				Truncated
Standard ML				Truncated
Stata				Euclidean
Swift				Truncated
Swift				Rounded
Swift				Truncated
Tcl				Floored
Tcl				Truncated
tcsh				Truncated
Torque				Truncated
Turing				Floored
Verilog				Truncated
VHDL				Floored
VHDL				Truncated
VimL				Truncated
Visual Basic				Truncated
WebAssembly	,
WebAssembly	,			Truncated
x86 assembly				Truncated
XBase++				Truncated
XBase++				Floored
Zig	,			Truncated
Zig				Floored
Z3 theorem prover	,			Euclidean

In addition, many computer systems provide a functionality, which produces the quotient and the remainder at the same time. Examples include the x86 architecture's instruction, the C programming language's function, and Python's function.