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This article is about the binary operation a mod m. For the (mod m) notation, see Modular arithmetic. For other uses, see Modulo (disambiguation).

In mathematics and computing, modulo is the binary operation of which the value is the remainder or signed remainder of a division of one number by another, the latter being called the modulus of the operation.

Given two positive numbers a and m, a modulo m (often written as $a mod m$ ) is the remainder of the Euclidean division of a by m, where a is the dividend and m is the divisor.^[1]

For example, $5 mod 2 = 1$ , because dividing 5 by 2 yields a remainder of 1 (besides the quotient of 2), while $9 mod 3 = 0$ , because 3 divides evenly into 9, thus leaving a remainder of 0.

In computing, although the modulo operation is typically applied with both of its arguments, a and m, being integers, many systems now allow other types of numeric operands. When they are both integers, the range of values for $a mod m$ is from 0 to $m - 1$ , inclusive. The value of $a mod 1$ is always 0, for any integer a. When exactly one of a or m is negative, the basic definition breaks down, and programming languages differ in how these values are defined.

Variants of the definition

In mathematics, the value of an expression $a mod m$ is, technically, 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 (i.e., the remainder of the Euclidean division).^[2] For example, $11 mod 3$ equals not 2 (the least positive residue), but the entire set ${..., - 4, - 1, 2, 5, 8, 11, 14, ...}$ of all integers that are congruent to 2 modulo 3 (which is to say those integers that are 2 more than a multiple of 3). This equivalence class would typically be denoted by $[2]$ .

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 the programming language or the underlying hardware.

In nearly all computing systems, the quotient q and the remainder r of a divided by $n \neq 0$ satisfy the following conditions:

\begin{matrix} q \in ℤ \\ a = n q + r (n \neq 0) \\ | r | < | n | \end{matrix}

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 of a or n.^[a] Standard Pascal and ALGOL 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 of n or a is negative (see the table under § In programming languages for details). Some systems leave a modulo 0 undefined, though others define it as a.

Quotient (q) and remainder (r) as functions of dividend (a), using truncated division

Many implementations use truncated division, for which the quotient is defined by $q = trunc (\frac{a}{n})$ where $trunc$ is the integral part function (rounding toward zero), i.e. the truncation to zero significant digits. Thus according to equation (1), the remainder has the same sign as the dividend a so can take 2|n| − 1 values: $r = a - n trunc (\frac{a}{n})$
Quotient and remainder using floored division

Donald Knuth^[3] promotes floored division, for which the quotient is defined by $q = ⌊ \frac{a}{n} ⌋$ where $⌊ ⌋$ is the floor function (rounding down). Thus according to equation (1), the remainder has the same sign as the divisor n: $r = a - n ⌊ \frac{a}{n} ⌋$
Quotient and remainder using Euclidean division

Raymond T. Boute^[4] promotes Euclidean division, for which the non-negative remainder $r \in {0, 1, 2...}$ is defined by $r := a - n q s u c h t h a t 0 \leq r < | n | .$ (Emphasis added.) Under this definition, we can say the following about the quotient $q$ : $\begin{matrix} q & = \frac{a - r}{n} \in ℤ \\ = sgn (n) \cdot \frac{a - r}{| n |} \\ = sgn (n) \cdot (\frac{a}{| n |} - \frac{r}{| n |}) \\ = sgn (n) \cdot ⌊ \frac{a}{| n |} ⌋ \end{matrix}$ where sgn is the sign function, $⌊ ⌋$ is the floor function (rounding down), and $\frac{a}{| n |} \in ℚ$ , $\frac{r}{| n |} \in ℚ$ are rational numbers.

Equivalently, one may instead define the quotient $q \in ℤ$ as follows: $q := sgn (n) ⌊ \frac{a}{| n |} ⌋ = {\begin{matrix} ⌊ \frac{a}{n} ⌋ & if n > 0 \\ ⌈ \frac{a}{n} ⌉ & if n < 0 \end{matrix}$ where $⌈ ⌉$ is the ceiling function (rounding up). Thus according to equation (1), the remainder $r$ is non-negative: $r = a - n q = a - | n | ⌊ \frac{a}{| n |} ⌋$
Quotient and remainder using rounded division

Common Lisp and IEEE 754 use rounded division, for which the quotient is defined by $q = round (\frac{a}{n})$ where round is the round 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: $r = a - n round (\frac{a}{n})$
Quotient and remainder using ceiling division

Common Lisp also uses ceiling division, for which the quotient is defined by $q = ⌈ \frac{a}{n} ⌉$ where ⌈⌉ is the ceiling function (rounding up). Thus according to equation (1), the remainder has the opposite sign of that of the divisor: $r = a - n ⌈ \frac{a}{n} ⌉$

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

This section is about the binary mod operation. For the (mod m) notation, see congruence relation.

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.

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 (truncated definition), 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(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 the remainder is not 0 (because remainder 0 is the same regardless of the signs):

bool is_odd(int n) {
    return n % 2 != 0;
}

Or with the binary arithmetic:

bool is_odd(int n) {
    return n & 1;
}

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 (assuming x 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 form expression % constant where constant is a power of two and automatically implement them as expression & (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 (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 & (constant-1) will always be positive. For these languages, the equivalence x % 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 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 a and n are integers.

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 (that is when b and n are coprime), and undefined otherwise.
Inverse multiplication: [(ab mod n)(b⁻¹ mod n)] mod n = a mod n.

Diophantine definition

For non-negative integers a, n and r, r = a mod n holds if and only if there are non-negative integers q and s with $a = n q + r and r + s + 1 = n,$ since the second equation expresses $r < n$ . The two equations can be combined into the single Diophantine equation $(a - n q - r)^{2} + (r + s + 1 - n)^{2} = 0,$ so the graph of the modulo operation on the non-negative integers is a Diophantine set. The same equation determines the quotient q, so the graph of divmod is Diophantine as well. The equation has no solution for n = 0, in line with leaving a modulo 0 undefined. Since the modulo operation is computable, its graph being Diophantine also follows from the MRDP theorem.

In programming languages

Modulo operators in various programming languages
Language	Operator	Integer	Floating-point	Definition
ABAP	`MOD`	Yes	Yes	Euclidean^[8]
ActionScript	`%`	Yes	Yes	Truncated^[9]
Ada	`mod`	Yes	No	Floored^[10]
Ada	`rem`	Yes	No	Truncated^[10]
ALGOL 68	`÷×`, `÷`, `%×`, `%`, `mod`^[b]	Yes	No	Euclidean^[11]
AMPL	`mod`	Yes	No	Truncated
APL	`\|`^[c]	Yes	Yes	Floored
AppleScript	`mod`	Yes	Yes	Truncated
AutoLISP	`rem`	Yes	Yes	Truncated
AWK	`%`	Yes	Yes	Truncated (same as `fmod` in C)^[12]
BASIC	`Mod`	Yes	No	Varies by implementation
bc	`%`	Yes	Yes	Truncated^[13]
C C++	`%`, `div`	Yes	No	Truncated^[d]
	`fmod` (C) `std::fmod` (C++)	No	Yes	Truncated^[16]
	`remainder` (C) `std::remainder` (C++)	No	Yes	Rounded^[17]
C#	`%`	Yes	Yes	Truncated^[18]
C#	`Math.IEEERemainder`	No	Yes	Rounded^[19]
Clarion	`%`	Yes	No	Truncated
Clean	`rem`	Yes	No	Truncated
Clojure	`mod`	Yes	No	Floored^[20]
Clojure	`rem`	Yes	No	Truncated^[21]
COBOL	`FUNCTION MOD`	Yes	No	Floored^[22]
COBOL	`FUNCTION REM`	Yes	Yes	Truncated^[22]
CoffeeScript	`%`	Yes	No	Truncated (same as JavaScript)^[23]
CoffeeScript	`%%`	Yes	No	Floored^[23]
ColdFusion	`%`, `MOD`	Yes	No	Truncated
Common Intermediate Language	`rem` (signed)	Yes	Yes	Truncated^[24]
Common Intermediate Language	`rem.un` (unsigned)	Yes	No	—N/a
Common Lisp	`mod`	Yes	Yes	Floored^[25]
Common Lisp	`rem`	Yes	Yes	Truncated^[25]
Crystal	`%`, `modulo`	Yes	Yes	Floored
Crystal	`remainder`	Yes	Yes	Truncated
CSS	`mod()`	Yes	Yes	Floored^[26]
CSS	`rem()`	Yes	Yes	Truncated^[27]
D	`%`	Yes	Yes	Truncated^[28]
Dart	`%`	Yes	Yes	Euclidean^[29]
Dart	`remainder()`	Yes	Yes	Truncated^[30]
dc	`%`	Yes	Yes	Truncated^[31]
Eiffel	`\\`	Yes	No	Truncated
Elixir	`rem/2`	Yes	No	Truncated^[32]
Elixir	`Integer.mod/2`	Yes	No	Floored^[33]
Elm	`modBy`	Yes	No	Floored^[34]
Elm	`remainderBy`	Yes	No	Truncated^[35]
Erlang	`rem`	Yes	No	Truncated
Erlang	`math:fmod/2`	No	Yes	Truncated (same as C)^[36]
Euphoria	`remainder`	Yes	Yes	Truncated^[37]
Euphoria	`mod`	Yes	Yes	Floored^[38]
F#	`%`	Yes	Yes	Truncated (same as C#)^[39]
F#	`Math.IEEERemainder`	No	Yes	Rounded^[19]
Factor	`mod`	Yes	Yes	Truncated^[40]
Factor	`rem`	Yes	Yes	Euclidean^[41]
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^[42]
Fortran	`modulo`	Yes	Yes	Floored^[43]
Frink	`%`, `mod`	Yes	Yes	Floored^[44]
Full BASIC	`MOD`	Yes	Yes	Floored^[45]
Full BASIC	`REMAINDER`	Yes	Yes	Truncated^[46]
GLSL	`%`	Yes	No	Undefined^[47]
GLSL	`mod`	No	Yes	Floored^[48]
GameMaker Studio (GML)	`mod`, `%`	Yes	No	Truncated
GDScript (Godot)	`%`	Yes	No	Truncated^[49]
	`posmod`	Yes	No	Euclidean^[50]
	`fmod`	No	Yes	Truncated^[51]
	`fposmod`	No	Yes	Euclidean^[52]
Go	`%`	Yes	No	Truncated^[53]
	`math.Mod`	No	Yes	Truncated^[54]
	`big.Int.Mod`	Yes	No	Euclidean^[55]
	`big.Int.Rem`	Yes	No	Truncated^[56]
Groovy	`%`	Yes	No	Truncated
Haskell	`mod`	Yes	No	Floored^[57]
	`rem`	Yes	No	Truncated^[57]
	`Data.Fixed.mod'` (GHC)	No	Yes	Floored
Haxe	`%`	Yes	Yes	Truncated^[58]
HLSL	`%`	Yes	Yes	Undefined^[59]
J	`\|`^[c]	Yes	No	Floored
Java	`%`	Yes	Yes	Truncated^[60]
	`Integer.remainderUnsigned`, `Long.remainderUnsigned`	Yes	No	—N/a^[61]^[62]
	`Math.ceilMod`, `StrictMath.ceilMod`	Yes	No	Ceiling^[63]^[64]^[65]^[66]^[67]^[68]
	`Math.floorMod`, `StrictMath.floorMod`	Yes	No	Floored^[69]^[70]^[71]^[72]^[73]^[74]
	`Math.IEEEremainder`, `StrictMath.IEEEremainder`	No	Yes	Rounded^[75]^[76]
	`a.mod(b)` (`BigInteger`)	Yes	No	Euclidean^[77]
	`a.remainder(b)` (`BigInteger`)	Yes	No	Truncated^[78]
	`a.remainder(b)`, `a.remainder(b, mathContext)` (`BigDecimal`)	No	Yes	Truncated^[79]^[80]
JavaScript TypeScript	`%`	Yes	Yes	Truncated
Julia	`mod`	Yes	Yes	Floored^[81]
Julia	`%`, `rem`	Yes	Yes	Truncated^[82]
Kotlin	`%`, `rem`	Yes	Yes	Truncated^[83]
Kotlin	`mod`	Yes	Yes	Floored^[84]
ksh	`%`	Yes	No	Truncated (same as POSIX `sh`)
ksh	`fmod`	No	Yes	Truncated
LabVIEW	`mod`	Yes	Yes	Truncated
LibreOffice	`MOD`	Yes	Yes	Floored^[85]
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	Yes	Floored^[86]
MATLAB	`mod`	Yes	Yes	Floored^[87]
MATLAB	`rem`	Yes	Yes	Truncated^[88]
Maxima	`mod`	Yes	No	Floored
Maxima	`remainder`	Yes	No	Truncated
Maya Embedded Language	`%`	Yes	No	Truncated
Maya Embedded Language	`fmod`	No	Yes	Truncated (same as C)^[89]
Microsoft Excel	`MOD`	Yes	Yes	Floored^[90]
Minitab	`MOD`	Yes	No	Floored
Modula-2	`MOD`	Yes	No	Floored
Modula-2	`REM`	Yes	No	Truncated
MUMPS	`#`	Yes	No	Floored^[91]
Netwide Assembler (NASM, NASMX)	`%`, `div` (unsigned)	Yes	No	—N/a
Netwide Assembler (NASM, NASMX)	`%%` (signed)	Yes	No	Implementation-defined^[92]
Nim	`mod`	Yes	Yes	Truncated^[93]^[94]
Nim	`floorMod`	Yes	Yes	Floored^[95]
Oberon	`MOD`	Yes	No	Floored-like^[e]
Objective-C	`%`	Yes	No	Truncated (same as C99)
Object Pascal, Delphi	`mod`	Yes	No	Truncated
OCaml	`mod`, `Int.rem`	Yes	No	Truncated^[96]^[97]
OCaml	`mod_float`, `Float.rem`	No	Yes	Truncated^[98]^[99]
Occam	`\`	Yes	No	Truncated
Pascal (ISO-7185 and -10206)	`mod`	Yes	No	Euclidean-like^[f]
Perl	`%`	Yes	No	Floored^[g]
Perl	`POSIX::fmod`	No	Yes	Truncated (same as C)^[101]
PHP	`%`	Yes	No	Truncated^[102]
PHP	`fmod`	No	Yes	Truncated^[103]
PIC BASIC 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	`mod`	Yes	No	Floored
Prolog	`rem`	Yes	No	Truncated
PureBasic	`%`	Yes	No	Truncated
PureBasic	`Mod(x,y)`	No	Yes	Truncated^[104]
PureScript	`mod`	Yes	No	Euclidean^[105]
Pure Data	`%`	Yes	No	Truncated (same as C)
Pure Data	`mod`	Yes	No	Floored
Python	`%`	Yes	Yes	Floored^[106]
	`math.fmod`	No	Yes	Truncated^[107]
	`math.remainder`	No	Yes	Rounded^[108]
Q#	`%`	Yes	No	Truncated^[109]
R	`%%`	Yes	Yes	Floored^[110]
Racket	`modulo`	Yes	Some^[h]	Floored^[112]
Racket	`remainder`	Yes	Some^[h]	Truncated^[113]
Raku	`%`	No	Yes	Floored
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^[114]
Rust	`rem_euclid()`	Yes	Yes	Euclidean^[115]
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^[116]
	`mod0`	Yes	No	Rounded^[116]
	`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) (includes bash, mksh, &c.)	`%`	Yes	No	Truncated (same as C)^[117]
Smalltalk	`\\`	Yes	Yes	Floored^[118]^{: sec. 5.6.2.9}
Smalltalk	`rem:`	Yes	Yes	Truncated^[118]^{: sec. 5.6.2.30}
Snap!	`mod`	Yes	No	Floored
Spin	`//`	Yes	No	Floored
Solidity	`%`	Yes	No	Truncated^[119]
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^[120]
	`remainder(dividingBy:)`	No	Yes	Rounded^[121]
	`truncatingRemainder(dividingBy:)`	No	Yes	Truncated^[122]
Tcl	`expr a % n`, `::tcl::mathop::% a n`^[i]	Yes	No	Floored^[124]^[125]
Tcl	`expr fmod(a, n)`, `::tcl::mathfunc::fmod a n`^[j]	No	Yes	Truncated (same 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	—N/a^[127]
WebAssembly	`i32.rem_s`, `i64.rem_s` (signed)	Yes	No	Truncated^[127]
x86 assembly	`DIV` (unsigned)	Yes	No	—N/a
	`IDIV` (signed)	Yes	No	Truncated
	`FPREM`	No	Yes	Truncated
	`FPREM1`	No	Yes	Rounded
XBase++	`%`	Yes	Yes	Truncated
XBase++	`Mod()`	Yes	Yes	Floored
Xojo	`Mod`	Yes	Partial^[k]	Truncated^[128]
Zig	`%`, `@rem`	Yes	Yes	Truncated^[129]
Zig	`@mod`	Yes	Yes	Floored
Z3 Theorem Prover	`div`, `mod`	Yes	No	Euclidean

In addition, many computer systems provide a divmod functionality, which produces the quotient and the remainder at the same time. Examples include the x86 architecture's DIV and IDIV instructions, the C programming language's div() function, and Python's divmod() function.

Generalizations

Modulo with offset

Sometimes it is useful for the result of a modulo n to lie not between 0 and n − 1, but between some number d and d + n − 1. In that case, d is called an offset and d = 1 is particularly common.

There does not seem to be a standard notation for this operation, so let us tentatively use a mod_d n. We thus have the following definition:^[130] x = a mod_d n just in case d ≤ x ≤ d + n − 1 and x mod n = a mod n. Clearly, the usual modulo operation corresponds to zero offset: a mod n = a mod₀ n.

The operation of modulo with offset is related to the floor function as follows: $a \mod_{d} n = a - n ⌊ \frac{a - d}{n} ⌋ .$

To see this, let $x = a - n ⌊ \frac{a - d}{n} ⌋$ . We first show that x mod n = a mod n. It is in general true that (a + bn) mod n = a mod n for all integers b; thus, this is true also in the particular case when $b = - ⌊ \frac{a - d}{n} ⌋$ ; but that means that $x mod n = (a - n ⌊ \frac{a - d}{n} ⌋) mod n = a mod n$ , which is what we wanted to prove. It remains to be shown that d ≤ x ≤ d + n − 1. Let k and r be the integers such that a − d = kn + r with 0 ≤ r ≤ n − 1 (see Euclidean division). Then $⌊ \frac{a - d}{n} ⌋ = k$ , thus $x = a - n ⌊ \frac{a - d}{n} ⌋ = a - n k = d + r$ . Now take 0 ≤ r ≤ n − 1 and add d to both sides, obtaining d ≤ d + r ≤ d + n − 1. But we've seen that x = d + r, so we are done.

The modulo with offset a mod_d n is implemented in Mathematica as Mod[a, n, d] .^[130]

Implementing other modulo definitions using truncation

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() */
typedef struct {
  /* This structure is part of the C stdlib.h, but is reproduced here for clarity */
  long int quot;
  long int rem;
} ldiv_t;

/* Euclidean division */
inline ldiv_t ldivE(long numer, long denom) {
  /* The C99 and C++11 languages define both of these as truncating. */
  long q = numer / denom;
  long r = 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 */
inline ldiv_t ldivF(long numer, long denom) {
  long q = numer / denom;
  long r = 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

↑ Mathematically, these two choices are but two of the infinite number of choices available for the inequality satisfied by a remainder.
↑ The revised report uses boldface to distinguish between keywords and identifiers, but permits other formats, including capitalization (e.g. MOD).
1 2 Argument order reverses, i.e., α|ω computes $ω mod α$ , the remainder when dividing ω by α.
↑ C99 and C++11 define the behavior of % to be truncated.^[14] The standards before then leave the behavior implementation-defined.^[15]
↑ Divisor must be positive, otherwise undefined.
↑ As discussed by Boute, ISO Pascal's definitions of div and mod do not obey the Division Identity of D = 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.^[100]
1 2 Racket defines an integer as a rational number for which round produces the same number when applied. This includes some inexact numbers, like 42.0, but excludes others, like 3.14.^[111]
↑ The expr command is defined independently of commands in the ::tcl::mathop namespace.^[123]
↑ The expr command is defined in terms of commands in the ::tcl::mathfunc namespace.^[126]
↑ Coerces floating-point operands to integers before calculating the result.

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External links

Modulorama, animation of a cyclic representation of multiplication tables (explanation in French)

Retrieved from "https://en.wikipedia.org/w/index.php?title=Modulo&oldid=1381283025"

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