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Modulo

From Wikipedia, the free encyclopedia

This article is about the binary operationmod(a,n). For the(modn) notation, seeModular arithmetic. For other uses, seeModulo (disambiguation).

Computational operation

Incomputing andmathematics, themodulo operation returns theremainder or signed remainder of adivision, after one number is divided by another, the latter being called themodulus of the operation.

Given two positive numbersa andn,a modulon (often abbreviated asa modn) is the remainder of theEuclidean division ofa byn, wherea is thedividend andn is thedivisor.^[1]

For example, the expression "5 mod 2" evaluates to 1, because 5 divided by 2 has aquotient 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 witha andn both beingintegers, many computing systems now allow other types of numeric operands. The range of values for an integer modulo operation ofn is 0 ton − 1.a mod 1 is always 0.

When exactly one ofa orn is negative, the basic definition breaks down, andprogramming languages differ in how these values are defined.

Variants of the definition

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Inmathematics, the result of themodulo operation is anequivalence class, and any member of the class may be chosen asrepresentative; however, the usual representative is theleast positive residue, the smallest non-negative integer that belongs to that class (i.e., the remainder of theEuclidean division).^[2] 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 theprogramming language or the underlyinghardware.

In nearly all computing systems, the quotientq and the remainderr ofa divided byn satisfy the following conditions:

{\begin{aligned}&q\in \mathbb {Z} \\&a=nq+r\\&|r|<|n|\end{aligned}}

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 (1). In number theory, the positive remainder is always chosen, but in computing, programming languages choose depending on the language and the signs ofa orn.^[a] StandardPascal andALGOL 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 ofn ora is negative (see the table under§ In programming languages for details). Some systems leavea modulo 0 undefined, though others define it asa.

Quotient (q) and remainder (r) as functions of dividend (a), using truncated division
Many implementations usetruncated division, for which the quotient is defined by
$q=\operatorname {trunc} \left({\frac {a}{n}}\right)$
where $\operatorname {trunc}$ is theintegral part function (rounding toward zero), i.e. thetruncation to zero significant digits.Thus according to equation (1), the remainder has thesame sign as the dividenda so can take2|n| − 1 values:
$r=a-n\operatorname {trunc} \left({\frac {a}{n}}\right)$
Quotient and remainder using floored division
Donald Knuth^[3] promotesfloored division, for which the quotient is defined by
$q=\left\lfloor {\frac {a}{n}}\right\rfloor$
where $\lfloor \,\rfloor$ is thefloor function (rounding down).Thus according to equation (1), the remainder has thesame sign as the divisorn:
$r=a-n\left\lfloor {\frac {a}{n}}\right\rfloor$
Quotient and remainder using Euclidean division
Raymond T. Boute^[4] promotesEuclidean division, for which the quotient is defined by
$q=\operatorname {sgn}(n)\left\lfloor {\frac {a}{\left|n\right|}}\right\rfloor ={\begin{cases}\left\lfloor {\frac {a}{n}}\right\rfloor &{\text{if }}n>0\\\left\lceil {\frac {a}{n}}\right\rceil &{\text{if }}n<0\\\end{cases}}$
wheresgn is thesign function, $\lfloor \,\rfloor$ is thefloor function (rounding down), and $\lceil \,\rceil$ is theceiling function (rounding up).Thus according to equation (1), the remainder isnon negative:
$r=a-|n|\left\lfloor {\frac {a}{\left|n\right|}}\right\rfloor$
Quotient and remainder using rounded division
Common Lisp andIEEE 754 userounded division, for which the quotient is defined by
$q=\operatorname {round} \left({\frac {a}{n}}\right)$
whereround is theround function (rounding half to even).Thus according to equation (1), the remainder falls between $-{\frac {n}{2}}$ and ${\frac {n}{2}}$ , and its sign depends on which side of zero it falls to be within these boundaries:
$r=a-n\operatorname {round} \left({\frac {a}{n}}\right)$
Quotient and remainder using ceiling division
Common Lisp also usesceiling division, for which the quotient is defined by
$q=\left\lceil {\frac {a}{n}}\right\rceil$
where ⌈⌉ is theceiling function (rounding up).Thus according to equation (1), the remainder has theopposite sign of that of the divisor:
$r=a-n\left\lceil {\frac {a}{n}}\right\rceil$

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 $({-a})/b={-(a/b)}=a/({-b})$ .^[5]^[6]

Notation

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This section is about the binarymod operation. For the(mod m) notation, seecongruence relation.

Some calculators have amod() function button, and many programming languages have a similar function, expressed asmod(a,n), for example. Some also support expressions that use "%", "mod", or "Mod" as a modulo or remainderoperator, such asa % n ora mod n.

For environments lacking a similar function, any of the three definitions above can be used.

Common pitfalls

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When the result of a modulo operation has the sign of the dividend (truncated definition), it can lead to surprising mistakes.

For example, to test if an integer isodd, one might be inclined to test if the remainder by 2 is equal to 1:

boolis_odd(intn){returnn%2==1;}

But in a language where modulo has the sign of the dividend, that is incorrect, because whenn (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):

boolis_odd(intn){returnn%2!=0;}

Or with the binary arithmetic:

boolis_odd(intn){returnn&1;}

Performance issues

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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 ofpowers of 2 can alternatively be expressed as abitwise AND operation (assumingx is a positive integer, or using a non-truncating definition):

x % 2ⁿ == x & (2ⁿ - 1)

Examples:

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]

Compiler optimizations may recognize expressions of the formexpression % constant whereconstant is a power of two and automatically implement them asexpression & (constant-1), 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 (includingC), unless the dividend is of anunsigned integer type. This is because, if the dividend is negative, the modulo will be negative, whereasexpression & (constant-1) will always be positive. For these languages, the equivalencex % 2ⁿ == x < 0 ? x | ~(2ⁿ - 1) : x & (2ⁿ - 1) 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 theconstant-divisor optimization.

Properties (identities)

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Some modulo operations can be factored or expanded similarly to other mathematical operations. This may be useful incryptography proofs, such as theDiffie–Hellman key exchange. The properties involving multiplication, division, and exponentiation generally require thata andn are integers.

Identity:
- (a modn) modn =a modn.
- n^x modn = 0 for all positive integer values ofx.
- Ifp is aprime number which is not adivisor ofb, thenab^p−1 modp =a modp, due toFermat's little theorem.
Inverse:
- [(−a modn) + (a modn)] modn = 0.
- b⁻¹ modn denotes themodular multiplicative inverse, which is definedif and only ifb andn arerelatively prime, which is the case when the left hand side is defined:[(b⁻¹ modn)(b modn)] modn = 1.
Distributive:
- (a +b) modn = [(a modn) + (b modn)] modn.
- ab modn = [(a modn)(b modn)] modn.
Division (definition):⁠a/b⁠ modn = [(a modn)(b⁻¹ modn)] modn, when the right hand side is defined (that is whenb andn arecoprime), and undefined otherwise.
Inverse multiplication:[(ab modn)(b⁻¹ modn)] modn =a modn.

In programming languages

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Modulo operators in various programming languages
Language	Operator	Integer	Floating-point	Definition
ABAP	`MOD`	Yes	Yes	Euclidean
ActionScript	`%`	Yes	No	Truncated
Ada	`mod`	Yes	No	Floored^[8]
Ada	`rem`	Yes	No	Truncated^[8]
ALGOL 68	`÷×`,`mod`	Yes	No	Euclidean
AMPL	`mod`	Yes	No	Truncated
APL	`\|`^[b]	Yes	Yes	Floored
AppleScript	`mod`	Yes	No	Truncated
AutoLISP	`(rem d n)`	Yes	No	Truncated
AWK	`%`	Yes	No	Truncated
bash	`%`	Yes	No	Truncated
BASIC	`Mod`	Yes	No	Varies by implementation
bc	`%`	Yes	No	Truncated
C C++	`%`,`div`	Yes	No	Truncated^[c]
	`fmod` (C) `std::fmod` (C++)	No	Yes	Truncated^[11]
	`remainder` (C) `std::remainder` (C++)	No	Yes	Rounded
C#	`%`	Yes	Yes	Truncated
C#	`Math.IEEERemainder`	No	Yes	Rounded^[12]
Clarion	`%`	Yes	No	Truncated
Clean	`rem`	Yes	No	Truncated
Clojure	`mod`	Yes	No	Floored^[13]
Clojure	`rem`	Yes	No	Truncated^[14]
COBOL	`FUNCTION MOD`	Yes	No	Floored^[15]
COBOL	`FUNCTION REM`	Yes	Yes	Truncated^[15]
CoffeeScript	`%`	Yes	No	Truncated
CoffeeScript	`%%`	Yes	No	Floored^[16]
ColdFusion	`%`,`MOD`	Yes	No	Truncated
Common Intermediate Language	`rem` (signed)	Yes	Yes	Truncated^[17]
Common Intermediate Language	`rem.un` (unsigned)	Yes	No	—
Common Lisp	`mod`	Yes	Yes	Floored
Common Lisp	`rem`	Yes	Yes	Truncated
Crystal	`%`,`modulo`	Yes	Yes	Floored
Crystal	`remainder`	Yes	Yes	Truncated
CSS	`mod()`	Yes	Yes	Floored^[18]
CSS	`rem()`	Yes	Yes	Truncated^[19]
D	`%`	Yes	Yes	Truncated^[20]
Dart	`%`	Yes	Yes	Euclidean^[21]
Dart	`remainder()`	Yes	Yes	Truncated^[22]
Eiffel	`\\`	Yes	No	Truncated
Elixir	`rem/2`	Yes	No	Truncated^[23]
Elixir	`Integer.mod/2`	Yes	No	Floored^[24]
Elm	`modBy`	Yes	No	Floored^[25]
Elm	`remainderBy`	Yes	No	Truncated^[26]
Erlang	`rem`	Yes	No	Truncated
Erlang	`math:fmod/2`	No	Yes	Truncated (same as C)^[27]
Euphoria	`mod`	Yes	No	Floored
Euphoria	`remainder`	Yes	No	Truncated
F#	`%`	Yes	Yes	Truncated
F#	`Math.IEEERemainder`	No	Yes	Rounded^[12]
Factor	`mod`	Yes	No	Truncated
FileMaker	`Mod`	Yes	No	Floored
Forth	`mod`	Yes	No	Implementation defined
	`fm/mod`	Yes	No	Floored
	`sm/rem`	Yes	No	Truncated
Fortran	`mod`	Yes	Yes	Truncated
Fortran	`modulo`	Yes	Yes	Floored
Frink	`mod`	Yes	No	Floored
Full BASIC	`MOD`	Yes	Yes	Floored^[28]
Full BASIC	`REMAINDER`	Yes	Yes	Truncated^[29]
GLSL	`%`	Yes	No	Undefined^[30]
GLSL	`mod`	No	Yes	Floored^[31]
GameMaker Studio (GML)	`mod`,`%`	Yes	No	Truncated
GDScript (Godot)	`%`	Yes	No	Truncated
	`fmod`	No	Yes	Truncated
	`posmod`	Yes	No	Euclidean
	`fposmod`	No	Yes	Euclidean
Go	`%`	Yes	No	Truncated^[32]
	`math.Mod`	No	Yes	Truncated^[33]
	`big.Int.Mod`	Yes	No	Euclidean^[34]
	`big.Int.Rem`	Yes	No	Truncated^[35]
Groovy	`%`	Yes	No	Truncated
Haskell	`mod`	Yes	No	Floored^[36]
	`rem`	Yes	No	Truncated^[36]
	`Data.Fixed.mod'` (GHC)	No	Yes	Floored
Haxe	`%`	Yes	No	Truncated
HLSL	`%`	Yes	Yes	Undefined^[37]
J	`\|`^[b]	Yes	No	Floored
Java	`%`	Yes	Yes	Truncated
Java	`Math.floorMod`	Yes	No	Floored
JavaScript TypeScript	`%`	Yes	Yes	Truncated
Julia	`mod`	Yes	Yes	Floored^[38]
Julia	`%`,`rem`	Yes	Yes	Truncated^[39]
Kotlin	`%`,`rem`	Yes	Yes	Truncated^[40]
Kotlin	`mod`	Yes	Yes	Floored^[41]
ksh	`%`	Yes	No	Truncated (same as POSIX sh)
ksh	`fmod`	No	Yes	Truncated
LabVIEW	`mod`	Yes	Yes	Truncated
LibreOffice	`=MOD()`	Yes	No	Floored
Logo	`MODULO`	Yes	No	Floored
Logo	`REMAINDER`	Yes	No	Truncated
Lua 5	`%`	Yes	Yes	Floored
Lua 4	`mod(x,y)`	Yes	Yes	Truncated
Liberty BASIC	`MOD`	Yes	No	Truncated
Mathcad	`mod(x,y)`	Yes	No	Floored
Maple	`e mod m` (by default),`modp(e, m)`	Yes	No	Euclidean
	`mods(e, m)`	Yes	No	Rounded
	`frem(e, m)`	Yes	Yes	Rounded
Mathematica	`Mod[a, b]`	Yes	No	Floored
MATLAB	`mod`	Yes	Yes	Floored
MATLAB	`rem`	Yes	Yes	Truncated
Maxima	`mod`	Yes	No	Floored
Maxima	`remainder`	Yes	No	Truncated
Maya Embedded Language	`%`	Yes	No	Truncated
Microsoft Excel	`=MOD()`	Yes	Yes	Floored
Minitab	`MOD`	Yes	No	Floored
Modula-2	`MOD`	Yes	No	Floored
Modula-2	`REM`	Yes	No	Truncated
MUMPS	`#`	Yes	No	Floored
Netwide Assembler (NASM,NASMX)	`%`,`div` (unsigned)	Yes	No	—
Netwide Assembler (NASM,NASMX)	`%%` (signed)	Yes	No	Implementation-defined^[42]
Nim	`mod`	Yes	No	Truncated
Oberon	`MOD`	Yes	No	Floored-like^[d]
Objective-C	`%`	Yes	No	Truncated (same as C99)
Object Pascal,Delphi	`mod`	Yes	No	Truncated
OCaml	`mod`	Yes	No	Truncated^[43]
OCaml	`mod_float`	No	Yes	Truncated^[44]
Occam	`\`	Yes	No	Truncated
Pascal (ISO-7185 and -10206)	`mod`	Yes	No	Euclidean-like^[e]
Perl	`%`	Yes	No	Floored^[f]
Perl	`POSIX::fmod`	No	Yes	Truncated
Phix	`mod`	Yes	No	Floored
Phix	`remainder`	Yes	No	Truncated
PHP	`%`	Yes	No	Truncated^[46]
PHP	`fmod`	No	Yes	Truncated^[47]
PICBASIC Pro	`\\`	Yes	No	Truncated
PL/I	`mod`	Yes	No	Floored (ANSI PL/I)
PowerShell	`%`	Yes	No	Truncated
Programming Code (PRC)	`MATH.OP - 'MOD; (\)'`	Yes	No	Undefined
Progress	`modulo`	Yes	No	Truncated
Prolog (ISO 1995)	`mod`	Yes	No	Floored
Prolog (ISO 1995)	`rem`	Yes	No	Truncated
PureBasic	`%`,`Mod(x,y)`	Yes	No	Truncated
PureScript	`mod`	Yes	No	Euclidean^[48]
Pure Data	`%`	Yes	No	Truncated (same as C)
Pure Data	`mod`	Yes	No	Floored
Python	`%`	Yes	Yes	Floored
	`math.fmod`	No	Yes	Truncated
	`math.remainder`	No	Yes	Rounded
Q#	`%`	Yes	No	Truncated^[49]
R	`%%`	Yes	Yes	Floored^[50]
Racket	`modulo`	Yes	No	Floored
Racket	`remainder`	Yes	No	Truncated
Raku	`%`	No	Yes	Floored
RealBasic	`MOD`	Yes	No	Truncated
Reason	`mod`	Yes	No	Truncated
Rexx	`//`	Yes	Yes	Truncated
RPG	`%REM`	Yes	No	Truncated
Ruby	`%`,`modulo()`	Yes	Yes	Floored
Ruby	`remainder()`	Yes	Yes	Truncated
Rust	`%`	Yes	Yes	Truncated
Rust	`rem_euclid()`	Yes	Yes	Euclidean^[51]
SAS	`MOD`	Yes	No	Truncated
Scala	`%`	Yes	Yes	Truncated
Scheme	`modulo`	Yes	No	Floored
Scheme	`remainder`	Yes	No	Truncated
Scheme R⁶RS	`mod`	Yes	No	Euclidean^[52]
	`mod0`	Yes	No	Rounded^[52]
	`flmod`	No	Yes	Euclidean
	`flmod0`	No	Yes	Rounded
Scratch	`mod`	Yes	Yes	Floored
Seed7	`mod`	Yes	Yes	Floored
Seed7	`rem`	Yes	Yes	Truncated
SenseTalk	`modulo`	Yes	No	Floored
SenseTalk	`rem`	Yes	No	Truncated
`sh` (POSIX) (includesbash,mksh, &c.)	`%`	Yes	No	Truncated (same as C)^[53]
Smalltalk	`\\`	Yes	No	Floored
Smalltalk	`rem:`	Yes	No	Truncated
Snap!	`mod`	Yes	No	Floored
Spin	`//`	Yes	No	Floored
Solidity	`%`	Yes	No	Truncated^[54]
SQL (SQL:1999)	`mod(x,y)`	Yes	No	Truncated
SQL (SQL:2011)	`%`	Yes	No	Truncated
Standard ML	`mod`	Yes	No	Floored
	`Int.rem`	Yes	No	Truncated
	`Real.rem`	No	Yes	Truncated
Stata	`mod(x,y)`	Yes	No	Euclidean
Swift	`%`	Yes	No	Truncated^[55]
	`remainder(dividingBy:)`	No	Yes	Rounded^[56]
	`truncatingRemainder(dividingBy:)`	No	Yes	Truncated^[57]
Tcl	`%`	Yes	No	Floored
Tcl	`fmod()`	No	Yes	Truncated (as C)
tcsh	`%`	Yes	No	Truncated
Torque	`%`	Yes	No	Truncated
Turing	`mod`	Yes	No	Floored
Verilog (2001)	`%`	Yes	No	Truncated
VHDL	`mod`	Yes	No	Floored
VHDL	`rem`	Yes	No	Truncated
VimL	`%`	Yes	No	Truncated
Visual Basic	`Mod`	Yes	No	Truncated
WebAssembly	`i32.rem_u`,`i64.rem_u` (unsigned)	Yes	No	—^[58]
WebAssembly	`i32.rem_s`,`i64.rem_s` (signed)	Yes	No	Truncated^[58]
x86 assembly	`IDIV`	Yes	No	Truncated
XBase++	`%`	Yes	Yes	Truncated
XBase++	`Mod()`	Yes	Yes	Floored
Zig	`%`,`@rem`	Yes	Yes	Truncated^[59]
Zig	`@mod`	Yes	Yes	Floored
Z3 theorem prover	`div`,`mod`	Yes	No	Euclidean

In addition, many computer systems provide adivmod functionality, which produces the quotient and the remainder at the same time. Examples include thex86 architecture'sIDIV instruction, the C programming language'sdiv() function, andPython'sdivmod() function.

Generalizations

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Modulo with offset

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Sometimes it is useful for the result ofa modulon to lie not between 0 andn − 1, but between some numberd andd +n − 1. In that case,d is called anoffset andd = 1 is particularly common.

There does not seem to be a standard notation for this operation, so let us tentatively usea mod_dn. We thus have the following definition:^[60]x =a mod_dn just in cased ≤x ≤d +n − 1 andx modn =a modn. Clearly, the usual modulo operation corresponds to zero offset:a modn =a mod₀n.

The operation of modulo with offset is related to thefloor function as follows:

a\operatorname {mod} _{d}n=a-n\left\lfloor {\frac {a-d}{n}}\right\rfloor .

To see this, let ${\textstyle x=a-n\left\lfloor {\frac {a-d}{n}}\right\rfloor }$ . We first show thatx modn =a modn. It is in general true that(a +bn) modn =a modn for all integersb; thus, this is true also in the particular case when ${\textstyle b=-\!\left\lfloor {\frac {a-d}{n}}\right\rfloor }$ ; but that means that ${\textstyle x{\bmod {n}}=\left(a-n\left\lfloor {\frac {a-d}{n}}\right\rfloor \right)\!{\bmod {n}}=a{\bmod {n}}}$ , which is what we wanted to prove. It remains to be shown thatd ≤x ≤d +n − 1. Letk andr be the integers such thata −d =kn +r with0 ≤r ≤n − 1 (seeEuclidean division). Then ${\textstyle \left\lfloor {\frac {a-d}{n}}\right\rfloor =k}$ , thus ${\textstyle x=a-n\left\lfloor {\frac {a-d}{n}}\right\rfloor =a-nk=d+r}$ . Now take0 ≤r ≤n − 1 and addd to both sides, obtainingd ≤d +r ≤d +n − 1. But we've seen thatx =d +r, so we are done.

The modulo with offseta mod_dn is implemented inMathematica asMod[a, n, d] .^[60]

Implementing other modulo definitions using truncation

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Despite the mathematical elegance of Knuth's floored division and Euclidean division, it is generally much more common to find a truncated division-based modulo in programming languages. Leijen provides the following algorithms for calculating the two divisions given a truncated integer division:

/* Euclidean and Floored divmod, in the style of C's ldiv() */typedefstruct{/* This structure is part of the C stdlib.h, but is reproduced here for clarity */longintquot;longintrem;}ldiv_t;/* Euclidean division */inlineldiv_tldivE(longnumer,longdenom){/* The C99 and C++11 languages define both of these as truncating. */longq=numer/denom;longr=numer%denom;if(r<0){if(denom>0){q=q-1;r=r+denom;}else{q=q+1;r=r-denom;}}return(ldiv_t){.quot=q,.rem=r};}/* Floored division */inlineldiv_tldivF(longnumer,longdenom){longq=numer/denom;longr=numer%denom;if((r>0&&denom<0)||(r<0&&denom>0)){q=q-1;r=r+denom;}return(ldiv_t){.quot=q,.rem=r};}

For both cases, the remainder can be calculated independently of the quotient, but not vice versa. The operations are combined here to save screen space, as the logical branches are the same.

Notes

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^Mathematically, these two choices are but two of the infinite number of choices available forthe inequality satisfied by a remainder.
^^a ^bArgument order reverses, i.e.,α|ω computes $\omega {\bmod {\alpha }}$ , the remainder when dividingω byα.
^C99 andC++11 define the behavior of% to be truncated.^[9] The standards before then leave the behavior implementation-defined.^[10]
^Divisor must be positive, otherwise undefined.
^As discussed by Boute, ISO Pascal's definitions ofdiv andmod do not obey the Division Identity ofD =d · (D /d) +D %d, and are thus fundamentally broken.
^Perl usually uses arithmetic modulo operator that is machine-independent. For examples and exceptions, see the Perl documentation on multiplicative operators.^[45]

References

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^Weisstein, Eric W."Congruence".Wolfram MathWorld. Retrieved2020-08-27.
^Caldwell, Chris."residue".Prime Glossary. RetrievedAugust 27, 2020.
^Knuth, Donald. E. (1972).The Art of Computer Programming. Addison-Wesley.
^Boute, Raymond T. (April 1992)."The Euclidean definition of the functions div and mod".ACM Transactions on Programming Languages and Systems.14 (2). ACM Press (New York, NY, USA):127–144.doi:10.1145/128861.128862.hdl:1854/LU-314490.S2CID 8321674.
^Peterson, Doctor (5 July 2001)."Mod Function and Negative Numbers".Math Forum - Ask Dr. Math. Archived fromthe original on 2019-10-22. Retrieved22 October 2019.
^"Ada 83 LRM, Sec 4.5: Operators and Expression Evaluation".archive.adaic.com. Retrieved2025-03-03.
^Horvath, Adam (July 5, 2012)."Faster division and modulo operation - the power of two".
^^a ^bISO/IEC 8652:2012 - Information technology — Programming languages — Ada.ISO,IEC. 2012. sec. 4.5.5 Multiplying Operators.
^"C99 specification (ISO/IEC 9899:TC2)"(PDF). 2005-05-06. sec. 6.5.5 Multiplicative operators. Retrieved16 August 2018.
^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 implementation-defined
^ISO/IEC 9899:1990: Programming languages – C.ISO,IEC. 1990. sec. 7.5.6.4.Thefmod function returns the valuex - i * y, for some integeri such that, ify is nonzero, the result has the same sign asx and magnitude less than the magnitude ofy.
^^a ^bdotnet-bot."Math.IEEERemainder(Double, Double) Method (System)".Microsoft Learn. Retrieved2022-10-04.
^"clojure.core - Clojure v1.10.3 API documentation".clojure.github.io. Retrieved2022-03-16.
^"clojure.core - Clojure v1.10.3 API documentation".clojure.github.io. Retrieved2022-03-16.
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