模除

商數 ( $q$ ) 和餘數 ( $r$ ) 作為被除數 ( $a$ ) 的函數時的圖像。左侧是除数为正的情况，右侧除数为负。从上至下分别使用了：向零取整、向下取整和欧几里得除法。

模除（英語：modulo 有时也称作 modulus），又稱模数、取模、取模運算等，它得出一个数除以另一个数的余数。给定两个正数：被除数 $a$ 和除数 $n$ ， $a\,\operatorname {modulo} \,n$ （通常缩写为 $a{\bmod {n}}$ ），得到的是使用欧几里得除法时的余数。 $a{\bmod {1}}$ 所得之数被称为 $a$ 的小数部份。

举个例子：计算表达式 $5{\bmod {2}}$ 得到 $1$ ，因为 $5\div 2$ 的商为 $2$ 而余数为 $1$ ；而 $9{\bmod {3}}$ 得到 $0$ ，因为 $9\div 3$ 的商为 $3$ 而余数为 $0$ ；做除法不能整除时得到是有理数，平常使用的计算器采用了有限个数位的十进制表示法，它以小数点分隔整数部份和小数部份，比如 $5\div 2=2.5$ 。

虽然通常情况下模除的被除数 $a$ 和除数 $n$ 都是整数，但许多计算系统允许其他类型的数值运算，比如对浮点数取模。在所有计算系统中当被除数 $a$ 和除数 $n$ 都是正数时，结果的余数 $r$ 的范围为 $0\leq r<n$ ；而 $a{\bmod {0}}$ 经常是未定义的，在编程语言里可能会导致除以零错误。当被除数 $a$ 或除数 $n$ 为负数时，不同的编程语言对结果有不同的处理。

各种定义

在数学中，取模运算的结果就是欧几里德除法的余数。当然也有许多其他的定义方式。计算机和计算器有许多种表示和储存数字的方法，因此在不同的硬件环境下、不同的编程语言中，取模运算有着不同的定义。

几乎所有的计算系统中， $a$ 除以 $n$ 得到商 $q$ 和余数 $r$ 均满足以下式子：

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

1

即使如此，当余数非 $0$ 时，余数的符号仍然是有歧义的：余数非 $0$ 时，它的符号有两种选择，一个是正号而另一个是负号^[a]。通常情况下，在数论中总是选择正余数。但在编程中，选择余数的符号依赖于编程语言和被除数 $a$ 或除数 $n$ 的符号。在编程语言所定义的整数模除中，ISO/IEC标准Pascal^[1]和ALGOL 68^[2]，在计算出的余数r为负数时，返回正数 $r+|n|$ 作为结果^[b]；另一些编程语言如ISO/IEC C90，当被除数 $a$ 或除数 $n$ 是负数时，C90标准并没有做具体的规定，而是留给编译器去定义并实现^[3]。在大多数系统中 $a{\bmod {0}}$ 是未定义的，虽然有些系统定义它就等于 $a$ 。更多详情参见后续章节表格。

很多取模的实现都使用了“截断除法”（英語：truncated division），商经由截尾函数来定义，商向零取整，结果等于普通除法所得的小数靠近 $0$ 方向的第一个整数：
$q=\operatorname {trunc} \left({\frac {a}{n}}\right)$

余数和被除数符号一致：

$r=a-n\operatorname {trunc} \left({\frac {a}{n}}\right)$
高德纳定义的“下取整除法”（英語：floored division）^[4]，商经由下取整函数来定义，商总是向负无穷取整，即使商已经是负数：
$q=\left\lfloor {\frac {a}{n}}\right\rfloor$

余数和除数符号一致：

$r=a-n\left\lfloor {\frac {a}{n}}\right\rfloor$
Raymond T. Boute使用的欧几里得除法定义中^[5]，要求满足 $0\leq r<|n|$ ，在这种情况下：
$q=\operatorname {sgn}(n)\left\lfloor {\frac {a}{\left|n\right|}}\right\rfloor ={\begin{cases}\left\lfloor {\frac {a}{n}}\right\rfloor &{\text{if }}n>0\\-\left\lfloor -{\frac {a}{n}}\right\rfloor &{\text{if }}n<0\\\end{cases}}$

这里的 $\operatorname {sgn}$ 是符号函数，余数总是非负数:

$r=a-|n|\left\lfloor {\frac {a}{\left|n\right|}}\right\rfloor ={\begin{cases}a-n\left\lfloor {\frac {a}{n}}\right\rfloor &{\text{if }}n>0\\a+n\left\lfloor -{\frac {a}{n}}\right\rfloor &{\text{if }}n<0\\\end{cases}}$
Common Lisp的round函数和IEEE 754（英语：IEEE 754-1985）使用“修约除法”（英語：rounded division），商经由修约函数 $\operatorname {round}$ （约半成偶）来定义为：
$q=\operatorname {round} \left({\frac {a}{n}}\right)$

当商为偶数时，余数范围是 ${\textstyle -{\frac {n}{2}}\leq r\leq {\frac {n}{2}}}$ ；当商为奇数时，余数范围是 ${\textstyle -{\frac {n}{2}}<r<{\frac {n}{2}}}$ ：

$r=a-n\operatorname {round} \left({\frac {a}{n}}\right)$
Common Lisp的ceiling函数使用“上取整除法”（英語：ceiling division），商经由上取整函数定义为：
$q=\left\lceil {\frac {a}{n}}\right\rceil$

余数与除数有相反的符号：

$r=a-n\left\lceil {\frac {a}{n}}\right\rceil$

记号

一些计算器有取模 $mod()$ 按钮，很多编程语言里也有类似的函数，通常像mod(a, n)这样。有些语言也支持在表达式内使用%、mod或Mod作为取模或取余操作符，比如a % n或a mod n。

在一些没有 $mod()$ 函数的环境中或许可使用等价的：a - (n * int(a/n))。这里的int()函数事实上等价于截断函数。

性质及恒等式

一些取模操作，经过分解和展开可以等同于其他数学运算。这在密码学的证明中十分有用，例如：迪菲-赫爾曼密鑰交換。

恒等式：
- $(a{\bmod {n}}){\bmod {n}}=a{\bmod {n}}$
- 对所有的正数 $x$ 有： $n^{x}{\bmod {n}}=0$
- 如果 $p$ 是一个质数，且不为 $b$ 的因数，此时由费马小定理有： $ab^{p-1}{\bmod {p}}=a{\bmod {p}}$
逆运算：
- $[(-a{\bmod {n}})+(a{\bmod {n}})]={\bmod {n}}=0$ .
- $b^{-1}{\bmod {n}}$ 表示模反元素。当且仅当 $b$ 与 $n$ 互质时，等式左侧有定义： $[(b^{-1}{\bmod {n}})(b{\bmod {n}})]{\bmod {n}}=1$ 。
分配律：
- $(a+b){\bmod {n}}=[(a{\bmod {n}})+(b{\bmod {n}})]{\bmod {n}}$
- $ab{\bmod {n}}=[(a{\bmod {n}})(b{\bmod {n}})]{\bmod {n}}$
除法定义：仅当式子右侧有定义时，即 $b$ 、 $n$ 互质时有： ${\frac {a}{b}}{\bmod {n}}=[(a{\bmod {n}})(b^{-1}{\bmod {n}})]{\bmod {n}}$ ，其他情况为未定义的。
乘法逆元： $[(ab{\bmod {n}})(b^{-1}{\bmod {n}})]{\bmod {n}}=a{\bmod {n}}$ .

编程语言实现

在各种编程语言中的模除运算
语言	算符	整数	浮点数	定义
ABAP	`MOD`	是	是	欧几里得式
ActionScript	`%`	是	否	截断
Ada	`mod`	是	否	下取整^[6]
Ada	`rem`	是	否	截断^[6]
ALGOL 68	`÷×`, `mod`	是	否	类似欧几里得式^[c]
AMPL	`mod`	是	否	截断
APL	`\|`^[d]	是	是	下取整
AppleScript	`mod`	是	否	截断
AutoLISP	`(rem d n)`	是	否	截断
AWK	`%`	是	否	截断
bash	`%`	是	否	截断
BASIC	`Mod`	是	否	实现各异
bc	`%`	是	否	截断
C C++	`%`, `div`	是	否	截断^[e]
	`fmod` (C) `std::fmod` (C++)	否	是	截断^[9]
	`remainder` (C) `std::remainder` (C++)	否	是	修约
C#	`%`	是	是	截断
C#	`Math.IEEERemainder`	否	是	修约^[10]
Clarion（英语：Clarion (programming language)）	`%`	是	否	截断
Clean	`rem`	是	否	截断
Clojure	`mod`	是	否	下取整^[11]
Clojure	`rem`	是	否	截断^[12]
COBOL	`FUNCTION MOD`	是	否	下取整^[13]
COBOL	`FUNCTION REM`	是	是	截断^[13]
CoffeeScript	`%`	是	否	截断
CoffeeScript	`%%`	是	否	下取整^[14]
ColdFusion	`%`, `MOD`	是	否	截断
Common Intermediate Language	`rem` (有符号)	是	是	截断^[15]
Common Intermediate Language	`rem.un` (无符号)	是	否	不適用
Common Lisp	`mod`	是	是	下取整
Common Lisp	`rem`	是	是	截断
Crystal（英语：Crystal (programming language)）	`%`, `modulo`	是	是	下取整
Crystal（英语：Crystal (programming language)）	`remainder`	是	是	截断
D	`%`	是	是	截断^[16]
Dart	`%`	是	是	欧几里得式^[17]
Dart	`remainder()`	是	是	截断^[18]
Eiffel	`\\`	是	否	截断
Elixir	`rem/2`	是	否	截断^[19]
Elixir	`Integer.mod/2`	是	否	下取整^[20]
Elm	`modBy`	是	否	下取整^[21]
Elm	`remainderBy`	是	否	截断^[22]
Erlang	`rem`	是	否	截断
Erlang	`math:fmod/2`	否	是	截断 (同于C)^[23]
Euphoria	`mod`	是	否	下取整
Euphoria	`remainder`	是	否	截断
F#	`%`	是	是	截断
F#	`Math.IEEERemainder`	否	是	修约^[10]
Factor	`mod`	是	否	截断
FileMaker	`Mod`	是	否	下取整
Forth	`mod`	是	否	实现定义
	`fm/mod`	是	否	下取整
	`sm/rem`	是	否	截断
Fortran	`mod`	是	是	截断
Fortran	`modulo`	是	是	下取整
Frink（英语：Frink (programming language)）	`mod`	是	否	下取整
Full BASIC（英语：Full BASIC）	`MOD`	是	是	下取整^[24]
Full BASIC（英语：Full BASIC）	`REMAINDER`	是	是	截断^[25]
GLSL	`%`	是	否	未定义^[26]
GLSL	`mod`	否	是	下取整^[27]
GameMaker Studio (GML)	`mod`, `%`	是	否	截断
GDScript (Godot)	`%`	是	否	截断
	`fmod`	否	是	截断
	`posmod`	是	否	欧几里得式
	`fposmod`	否	是	欧几里得式
Go	`%`	是	否	截断^[28]
	`math.Mod`	否	是	截断^[29]
	`big.Int.Mod`	是	否	欧几里得式^[30]
	`big.Int.Rem`	是	否	截断^[31]
Groovy	`%`	是	否	截断
Haskell	`mod`	是	否	下取整^[32]
	`rem`	是	否	截断^[32]
	`Data.Fixed.mod'` (GHC)	否	是	下取整
Haxe	`%`	是	否	截断
HLSL	`%`	是	是	未定义^[33]
J	`\|`^[d]	是	否	下取整
Java	`%`	是	是	截断
Java	`Math.floorMod`	是	否	下取整
JavaScript TypeScript	`%`	是	是	截断
Julia	`mod`	是	是	下取整^[34]
Julia	`%`, `rem`	是	是	截断^[35]
Kotlin	`%`, `rem`	是	是	截断^[36]
Kotlin	`mod`	是	是	下取整^[37]
ksh	`%`	是	否	截断 (同于POSIX sh)
ksh	`fmod`	否	是	截断
LabVIEW	`mod`	是	是	截断
LibreOffice	`=MOD()`	是	否	下取整
Logo	`MODULO`	是	否	下取整
Logo	`REMAINDER`	是	否	截断
Lua 5	`%`	是	是	下取整
Lua 4	`mod(x,y)`	是	是	截断
Liberty BASIC（英语：Liberty BASIC）	`MOD`	是	否	截断
Mathcad	`mod(x,y)`	是	否	下取整
Maple	`e mod m` (默认), `modp(e, m)`	是	否	欧几里得式
	`mods(e, m)`	是	否	修约
	`frem(e, m)`	是	是	修约
Mathematica	`Mod[a, b]`	是	否	下取整
MATLAB	`mod`	是	否	下取整
MATLAB	`rem`	是	否	截断
Maxima	`mod`	是	否	下取整
Maxima	`remainder`	是	否	截断
Maya Embedded Language（英语：Maya Embedded Language）	`%`	是	否	截断
Microsoft Excel	`=MOD()`	是	是	下取整
Minitab	`MOD`	是	否	下取整
Modula-2	`MOD`	是	否	下取整
Modula-2	`REM`	是	否	截断
MUMPS（英语：MUMPS）	`#`	是	否	下取整
Netwide Assembler (NASM, NASMX)	`%`, `div` (无符号)	是	否	不適用
Netwide Assembler (NASM, NASMX)	`%%` (有符号)	是	否	实现定义^[38]
Nim	`mod`	是	否	截断
Oberon	`MOD`	是	否	类似下取整^[f]
Objective-C	`%`	是	否	截断 (同于C99)
Object Pascal, Delphi	`mod`	是	否	截断
OCaml	`mod`	是	否	截断^[39]
OCaml	`mod_float`	否	是	截断^[40]
Occam	`\`	是	否	截断
Pascal (ISO-7185和ISO-10206)	`mod`	是	否	类似欧几里得式^[c]
Perl	`%`	是	否	下取整^[g]
Perl	`POSIX::fmod`	否	是	截断
PHP	`%`	是	否	截断^[42]
PHP	`fmod`	否	是	截断^[43]
PIC BASIC Pro	`\\`	是	否	截断
PL/I	`mod`	是	否	下取整 (ANSI PL/I)
PowerShell	`%`	是	否	截断
Programming Code (PRC（英语：PRC (Palm OS)）)	`MATH.OP - 'MOD; (\)'`	是	否	未定义
Progress（英语：OpenEdge Advanced Business Language）	`modulo`	是	否	截断
Prolog (ISO 1995^[44])	`mod`	是	否	下取整
Prolog (ISO 1995^[44])	`rem`	是	否	截断
PureBasic	`%`, `Mod(x,y)`	是	否	截断
PureScript	`mod`	是	否	欧几里得式^[45]
Pure Data	`%`	是	否	截断 (同于C)
Pure Data	`mod`	是	否	下取整
Python	`%`	是	是	下取整
	`math.fmod`	否	是	截断
	`math.remainder`	否	是	修约
Q#	`%`	是	否	截断^[46]
R	`%%`	是	是	下取整^[47]
Racket	`modulo`	是	否	下取整
Racket	`remainder`	是	否	截断
Raku	`%`	否	是	下取整
RealBasic（英语：RealBasic）	`MOD`	是	否	截断
Reason	`mod`	是	否	截断
Rexx	`//`	是	是	截断
RPG（英语：IBM RPG）	`%REM`	是	否	截断
Ruby	`%`, `modulo()`	是	是	下取整
Ruby	`remainder()`	是	是	截断
Rust	`%`	是	是	截断
Rust	`rem_euclid()`	是	是	欧几里得式^[48]
SAS	`MOD`	是	否	截断
Scala	`%`	是	是	截断
Scheme	`modulo`	是	否	下取整
Scheme	`remainder`	是	否	截断
Scheme R⁶RS	`mod`	是	否	欧几里得式^[49]
	`mod0`	是	否	修约^[49]
	`flmod`	否	是	欧几里得式
	`flmod0`	否	是	修约
Scratch	`mod`	是	是	下取整
Seed7（英语：Seed7）	`mod`	是	是	下取整
Seed7（英语：Seed7）	`rem`	是	是	截断
SenseTalk（英语：SenseTalk）	`modulo`	是	否	下取整
SenseTalk（英语：SenseTalk）	`rem`	是	否	截断
Unix shell (包括bash、ash等。)	`%`	是	否	截断 (同于C)^[50]
Smalltalk	`\\`	是	否	下取整
Smalltalk	`rem:`	是	否	截断
Snap!	`mod`	是	否	下取整
Spin（英语：Parallax Propeller#Built in Spin byte code interpreter）	`//`	是	否	下取整
Solidity	`%`	是	否	下取整
SQL (SQL:1999（英语：SQL:1999）)	`mod(x,y)`	是	否	截断
SQL (SQL:2011（英语：SQL:2011）)	`%`	是	否	截断
Standard ML	`mod`	是	否	下取整
	`Int.rem`	是	否	截断
	`Real.rem`	否	是	截断
Stata	`mod(x,y)`	是	否	欧几里得式
Swift	`%`	是	否	截断^[51]
	`remainder(dividingBy:)`	否	是	修约^[52]
	`truncatingRemainder(dividingBy:)`	否	是	截断^[53]
Tcl	`%`	是	否	下取整
Tcl	`fmod()`	否	是	截断 (同于C)
tcsh	`%`	是	否	截断
Torque（英语：Torque (game engine)）	`%`	是	否	截断
Turing（英语：Turing (programming language)）	`mod`	是	否	下取整
Verilog (2001)	`%`	是	否	截断
VHDL	`mod`	是	否	下取整
VHDL	`rem`	是	否	截断
VimL	`%`	是	否	截断
Visual Basic	`Mod`	是	否	截断
WebAssembly	`i32.rem_u`, `i64.rem_u` (无符号)	是	否	不適用
WebAssembly	`i32.rem_s`, `i64.rem_s` (有符号)	是	否	截断^[54]
x86 assembly（英语：x86 assembly language）	`DIV`(无符号)	是	否	不適用
x86 assembly（英语：x86 assembly language）	`IDIV`(有符号)	是	否	截断
XBase++（英语：XBase++）	`%`	是	是	截断
XBase++（英语：XBase++）	`Mod()`	是	是	下取整
Zig	`%`, `@mod`, `@rem`	是	是	截断^[55]
Z3 theorem prover（英语：Z3 Theorem Prover）	`div`, `mod`	是	否	欧几里得式

此外，很多计算机系统提供divmod功能，它同时产生商和余数。例子x86架构的IDIV指令，C编程语言的div()函数，和Python的divmod()函数。

常见错误

当取模的结果与被除数符号相同时，可能会导致意想不到的错误。

举个例子：如果需要判断一个整数是否为奇数，有人可能会测试这个数除 2 的余数是否为 1：

bool is_odd(int n) {
    return n % 2 == 1;
}

但在一个取模结果与被除数符号相同的编程语言里，这样做是错的。因为当被除数 $n$ 是奇数且为负数时， $n{\bmod {2}}$ 得到 −1，此时函数返回“假”。

一种正确的实现是测试取模结果是否为 0，因为余数为 0 时没有符号的问题：

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

或者考虑余数的符号，有两种情况：余数可能为 1 或 -1。

bool is_odd(int n) {
    return n % 2 == 1 || n % 2 == -1;
}

性能问题

可以通过依次计算带余数的除法实现取模操作。特殊情况下，如某些硬件上，存在更快的实现。例如：2 的 n 次幂的模，可以通过逐位与运算实现：

x % 2ⁿ == x & (2ⁿ - 1)

例子，假定 $x$ 为正数：

x % 2 == x & 1

x % 4 == x & 3

x % 8 == x & 7

在进行位操作比取模操作效率更高的设备或软件环境中，以上形式的取模运算速度更快。^[56]

编译器可以自动识别出对 2 的 n 次幂取模的表达式，自动将其优化为 expression & (constant-1)。这样可以在兼顾效率的情况下写出更整洁的代码。这个优化在取模结果与被除数符号一致的语言中（包括 C 语言）不能使用，除非被除数是无符号整数。这是因为如果被除数是负数，则结果也是负数，但 expression & (constant-1) 总是正数，进行这样的优化就会导致错误，无符号整数则没有这个问题。

用途

取模运算可用于判断一个数是否能被另一个数整除。对 2 取模即可判断整数的奇偶性；从 2 到 $n-1$ 取模则可判断一个数是否为质数。
進制之間的轉換。
用于求取最大公约数的輾轉相除法使用取模运算。
密码学中的应用：从古老的凯撒密码到现代常用的RSA、椭圆曲线密码，它们的实现过程均使用了取模运算。

參見

模 (消歧义)和模 (术语)（英语：modulo (jargon)） —— “模数（Modulo）”这个词的许多用法，都是 1801 年卡爾·弗里德里希·高斯引入模算數时产生的。
模幂运算
同餘

脚注

^ 从数学上讲，正和负只是满足不等式的无穷多个解中的两个。
^ ISO Pascal要求模除运算的除数n为正数。
^ ^3.0 ^3.1 按Boute的研讨，ISO Pascal和Algol68的div和mod定义不能维持除法恒等式 $D = d \cdot (D / d) + D % d$ 。
^ ^4.0 ^4.1 α|ω计算 $\omega {\bmod {\alpha }}$ ，得出ω除以α的余数。
^ C99和C++11将%的行为定义为截断。^[7]此前的标准将其行为留给实现定义。^[8]
^ 除数必须是整数，否则未定义。
^ Perl通常使用机器无关的算术模除运算。其例子和异常，参见关于乘法运算的Perl文档。^[41]

参考文献

^ ISO/IEC 7185, Information Technology—Programming Languages–Pascal. International Standard, 2nd ed. (PDF). ISO/IEC 7185, 1990 (E).
A term of the form i div j shall be an error if j is zero; otherwise, the value of i div j shall be such that

abs(i) - abs(j) < abs((i div j) * j) <= abs(i)
where the value shall be zero if abs(i) < abs(j); otherwise, the sign of the value shall be positive if i and j have the same sign and negative if i and j have different signs.
A term of the form i mod j shall be an error if j is zero or negative; otherwise, the value of i mod j shall be that value of (i - (k * j)) for integral k such that 0 <= i mod j < j.
NOTE 2 — Only for i >= 0 and j > 0 does the relation (i div j) * j + i mod j = i hold.
Kathleen Jensen, Niklaus Wirth. PASCAL User Manual and Report － ISO Pascal Standard (PDF). 1991.

div   divide and truncate (i.e., value is not rounded)

mod   modulus: let Remainder = A - (A div B) * B;
  if Remainder < 0 then A mod B = Remainder + B

  otherwise A mod B = Remainder

^ A. van Wijngaarden, B. J. Mailloux, J. E. L. Peck, C. H. A. Koster, M. Sintzoff, C. H. Lindsey, L. G. L.T. Meertens and R. G. Fisker. Revised Report on the Algorithmic Language Algol 68. IFIP W.G. 2.1.
10.2.3.3. Operations on integral operands
……

m) OP «÷, %, OVER» = (L INT a, b) L INT:
      IF b ≠ L 0
      THEN L INT q := L 0, r := ABS a;
         WHILE (r := r - ABS b) ≥ L 0 DO q := q + L 1 OD;
         (a < L 0 AND b > L 0 OR a ≥ L 0 AND b < L 0 | -q | q)
      FI;
n) OP «÷×, ÷*, %*, MOD» = (L INT a, b) L INT:
      (L INT r = a - a ÷ b × b; r < 0 | r + ABS b | r);

^
Jones, Derek M. The New C Standard: An Economic and Cultural Commentary (PDF). Addison-Wesley. 2003 [2018-07-11]. ISBN 9780201709179. （原始内容 (PDF)存档于2018-07-11）（英语）.
C99 reflects almost universal processor behavior (as does the Fortran Standard). This definition truncates toward zero and the expression (-(a/b) == (-a)/b) && (-(a/b) == a/(-b)) is always true. It also means that the absolute value of the result does not depend on the signs of the operands; for example:

+10 / +3 == +3 +10 % +3 == +1 -10 / +3 == -3 -10 % +3 == -1
+10 / -3 == -3 +10 % -3 == +1 -10 / -3 == +3 -10 % -3 == -1
C90

When integers are divided and the division is inexact, if both operands are positive the result of the / operator is the largest integer less than the algebraic quotient and the result of the % operator is positive. If either operand is negative, whether the result of the / operator is the largest integer less than or equal to the algebraic quotient or the smallest integer greater than or equal to the algebraic quotient is implementation-defined, as is the sign of the result of the % operator.
If either operand is negative, the behavior may differ between C90 and C99, depending on the implementation-defined behavior of the C90 implementation.
#include <stdio.h> int main(void) { int x = -1, y = +3; if ((x%y > 0) || ((x+y)%y == x%y)) printf("This is a C90 translator behaving differently than C99\n"); }
Quoting from the C9X Revision Proposal, WG14/N613, that proposed this change:

The origin of this practice seems to have been a desire to map C’s division directly to the “natural” behavior of the target instruction set, whatever it may be, without requiring extra code overhead that might be necessary to check for special cases and enforce a particular behavior. However, the argument that Fortran programmers are unpleasantly surprised by this aspect of C and that there would be negligible impact on code efficiency was accepted by WG14, who agreed to require Fortran-like behavior in C99.

^ Knuth, Donald. E. The Art of Computer Programming. Addison-Wesley. 1972 （英语）.
If

x

is any real number, we write

\left\lfloor x\right\rfloor

= the greatest integer less than or equal to

x

(the floor of

x

);

\left\lceil x\right\rceil

= the least integer greater than or equal to

x

(the ceiling of

x

).

……
The following formulas and examples are easily verified:
……

$\left\lceil x\right\rceil =\left\lfloor x\right\rfloor$	if and only if $x$ is an integer,
$\left\lceil x\right\rceil =\left\lfloor x\right\rfloor +1$	if and only if $x$ is not an integer;
$\left\lfloor -x\right\rfloor =-\left\lceil x\right\rceil$ ;	$x-1<\left\lfloor x\right\rfloor \leq x\leq \left\lceil x\right\rceil <x+1$ .

……
If $x$ and $y$ are any real numbers, we define the following binary operation:

x{\bmod {y}}=x-y\left\lfloor x/y\right\rfloor ,\ {\text{if }}y\neq 0

;

x{\bmod {0}}=x

. (1)

From this definition we can see that, when $y\neq 0$ ,

0\leq {\frac {x}{y}}-\left\lfloor {\frac {x}{y}}\right\rfloor ={\frac {x{\bmod {y}}}{y}}<1

. (2)

Consequently
a) if $y>0$ , then $0\leq x{\bmod {y}}<y$ ;
b) if $y<0$ , then $0\geq x{\bmod {y}}>y$ ;
c) the quantity $x-(x{\bmod {y}})$ is an integral multiple of $y$ .
We call $x{\bmod {y}}$ the remainder when $x$ is divided by $y$ ; similarly, we call $\left\lfloor x/y\right\rfloor$ the quotient.
When $x$ and $y$ are integers, “ ${\text{mod}}$ ” is therefore a familiar operation:

5{\bmod {3}}=2

,

18{\bmod {3}}=0

,

-2{\bmod {3}}=1

. (3)

We have $x{\bmod {y}}=0$ if and only if $x$ is a multiple of $y$ , that is, if and only if $x$ is divisible by $y$ . The notation $y\backslash x$ , read “ $y$ divides $x$ ,” means that $y$ is a positive integer and $x{\bmod {y}}=0$ .
The “ ${\text{mod}}$ ” operation is useful also when $x$ and $y$ take arbitrary real values. For example, with trigonometric functions we can write

\tan x=\tan(x{\bmod {\pi }})

.

The quantity $x{\bmod {1}}$ is the fractional part of $x$ ; we have, by Eq. (1),

x=\left\lfloor x\right\rfloor +(x{\bmod {1}})

.

^ Boute, Raymond T. The Euclidean definition of the functions div and mod. ACM Transactions on Programming Languages and Systems (ACM Press (New York, NY, USA)). April 1992, 14 (2): 127–144. doi:10.1145/128861.128862 （英语）.
Let us recall Euclid’s theorem.

THEOREM. For any real numbers $D$ and $d$ with $d\neq 0$ , there exists a unique pair of numbers $q,r$ satisfying the following conditions:

(a) $q\in \mathbb {Z}$ ;

(b) $D=d\cdot q+r$ ;

(c) $0\leq r<|d|$ .
For the particular case of integers, the conditions (a)-(c) correspond to part of the axioms for Euclidean rings [9].
With the notational conventions of the theorem, we define $D\operatorname {div} d=q$ and $D{\bmod {d}}=r$ . Clearly the correspondingly labeled conditions are satisfied.
Properties

${\begin{aligned}D\operatorname {div} \,(-d)&=-(D\operatorname {div} d)\\D{\bmod {\left(-d\right)}}&=D{\bmod {d}}\\(-D)\operatorname {div} d&=-(D\operatorname {div} d)-I\cdot J\\(-D){\bmod {d}}&=-(D{\bmod {d}})+d\cdot I\cdot J\end{aligned}}$
where $J=\operatorname {sign} d$ and $I$ is as defined earlier. ( $I={\text{if }}D/d\in \mathbb {Z} {\text{ then }}0{\text{ else }}1$ )
^ ^6.0 ^6.1 ISO/IEC 8652:2012 - Information technology — Programming languages — Ada. ISO, IEC. 2012. sec. 4.5.5 Multiplying Operators.
^ C99 specification (ISO/IEC 9899:TC2) (PDF). sec. 6.5.5 Multiplicative operators. 2005-05-06 [16 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. The $fmod$ function returns the value $x - i * y$ , for some integer $i$ such that, if $y$ is nonzero, the result has the same sign as $x$ and magnitude less than the magnitude of $y$ .
^ ^10.0 ^10.1 dotnet-bot. Math.IEEERemainder(Double, Double) Method (System). Microsoft Learn. [2022-10-04] （美国英语）.
^ clojure.core - Clojure v1.10.3 API documentation. clojure.github.io. [2022-03-16].
^ clojure.core - Clojure v1.10.3 API documentation. clojure.github.io. [2022-03-16].
^ ^13.0 ^13.1 ISO/IEC JTC 1/SC 22/WG 4. ISO/IEC 1989:2023 – Programming language COBOL. ISO. January 2023.
^ CoffeeScript operators
^ ISO/IEC JTC 1/SC 22. ISO/IEC 23271:2012 — Information technology — Common Language Infrastructure (CLI). ISO. February 2012. §§ III.3.55–56.
^ Expressions - D Programming Language. dlang.org. [2021-06-01].
^ operator % method - num class - dart:core library - Dart API. api.dart.dev. [2021-06-01].
^ remainder method - num class - dart:core library - Dart API. api.dart.dev. [2021-06-01].
^ Kernel — Elixir v1.11.3. hexdocs.pm. [2021-01-28].
^ Integer — Elixir v1.11.3. hexdocs.pm. [2021-01-28].
^ Basics - core 1.0.5. package.elm-lang.org. [2022-03-16].
^ Basics - core 1.0.5. package.elm-lang.org. [2022-03-16].
^ Erlang -- math. erlang.org. [2021-06-01].
^ ANSI. Programming Languages — Full BASIC. New York: American National Standards Institute. 28 January 1987. § 5.4.4. X modulo Y, i.e., X-Y*INT(X/Y).
^ ANSI. Programming Languages — Full BASIC. New York: American National Standards Institute. 28 January 1987. § 5.4.4. The remainder function, i.e., X-Y*IP(X/Y).
^ GLSL Language Specification, Version 4.50.7 (PDF). section 5.9 Expressions. If both operands are non-negative, then the remainder is non-negative. Results are undefined if one or both operands are negative.
^ GLSL Language Specification, Version 4.50.7 (PDF). section 8.3 Common Functions.
^ The Go Programming Language Specification - The Go Programming Language. go.dev. [2022-02-28].
^ math package - math - pkg.go.dev. pkg.go.dev. [2022-02-28].
^ big package - math/big - pkg.go.dev. pkg.go.dev. [2022-02-28].
^ big package - math/big - pkg.go.dev. pkg.go.dev. [2024-04-12].
^ ^32.0 ^32.1 6 Predefined Types and Classes. www.haskell.org. [2022-05-22].
^ Operators. Microsoft. [2021-07-19]. The % operator is defined only in cases where either both sides are positive or both sides are negative. Unlike C, it also operates on floating-point data types, as well as integers.
^ Mathematics · The Julia Language. docs.julialang.org. [2021-11-20].
^ Mathematics · The Julia Language. docs.julialang.org. [2021-11-20].
^ rem - Kotlin Programming Language. Kotlin. [2021-05-05] （英语）.
^ mod - Kotlin Programming Language. Kotlin. [2021-05-05] （英语）.
^ Chapter 3: The NASM Language. NASM - The Netwide Assembler version 2.15.05.
^ OCaml library : Stdlib. ocaml.org. [2022-02-19].
^ OCaml library : Stdlib. ocaml.org. [2022-02-19].
^ Perl documentation
^ PHP: Arithmetic Operators - Manual. www.php.net. [2021-11-20].
^ PHP: fmod - Manual. www.php.net. [2021-11-20].
^ ISO 1995
^ EuclideanRing.
^ QuantumWriter. Expressions. docs.microsoft.com. [2018-07-11] （美国英语）.
^ R: Arithmetic Operators. search.r-project.org. [2022-12-24].
^ F32 - Rust.
^ ^49.0 ^49.1 r6rs.org
^ Shell Command Language. pubs.opengroup.org. [2021-02-05].
^ Apple Developer Documentation. developer.apple.com. [2021-11-20].
^ Apple Developer Documentation. developer.apple.com. [2021-11-20].
^ Apple Developer Documentation. developer.apple.com. [2021-11-20].
^ Rossberg, Andreas (编). WebAssembly Core Specification: Version 2.0. World Wide Web Consortium. § 4.3.2 Integer Operations. 19 April 2022.
^ Zig Documentation. Zig Programming Language. [2022-12-18].
^ Horvath, Adam. Faster division and modulo operation - the power of two. July 5, 2012. （原始内容存档于2018-03-05）（英语）.

[1] 从数学上讲，正和负只是满足不等式的无穷多个解中的两个。

[4] ISO Pascal要求模除运算的除数n为正数。

[pascal&algol68-9] 3.0 ^3.1 按Boute的研讨，ISO Pascal和Algol68的div和mod定义不能维持除法恒等式 $D = d \cdot (D / d) + D % d$ 。

[rev-10] 4.0 ^4.1 α|ω计算 $\omega {\bmod {\alpha }}$ ，得出ω除以α的余数。

[c-13] C99和C++11将%的行为定义为截断。^[7]此前的标准将其行为留给实现定义。^[8]

[44] 除数必须是整数，否则未定义。

[48] Perl通常使用机器无关的算术模除运算。其例子和异常，参见关于乘法运算的Perl文档。^[41]

[2] ISO/IEC 7185, Information Technology—Programming Languages–Pascal. International Standard, 2nd ed. (PDF). ISO/IEC 7185, 1990 (E).
A term of the form i div j shall be an error if j is zero; otherwise, the value of i div j shall be such that

abs(i) - abs(j) < abs((i div j) * j) <= abs(i)
where the value shall be zero if abs(i) < abs(j); otherwise, the sign of the value shall be positive if i and j have the same sign and negative if i and j have different signs.
A term of the form i mod j shall be an error if j is zero or negative; otherwise, the value of i mod j shall be that value of (i - (k * j)) for integral k such that 0 <= i mod j < j.
NOTE 2 — Only for i >= 0 and j > 0 does the relation (i div j) * j + i mod j = i hold.
Kathleen Jensen, Niklaus Wirth. PASCAL User Manual and Report － ISO Pascal Standard (PDF). 1991.

div divide and truncate (i.e., value is not rounded)

mod modulus: let Remainder = A - (A div B) * B;
if Remainder < 0 then A mod B = Remainder + B

otherwise A mod B = Remainder

[3] A. van Wijngaarden, B. J. Mailloux, J. E. L. Peck, C. H. A. Koster, M. Sintzoff, C. H. Lindsey, L. G. L.T. Meertens and R. G. Fisker. Revised Report on the Algorithmic Language Algol 68. IFIP W.G. 2.1.
10.2.3.3. Operations on integral operands
……

m) OP «÷, %, OVER» = (L INT a, b) L INT: IF b ≠ L 0 THEN L INT q := L 0, r := ABS a; WHILE (r := r - ABS b) ≥ L 0 DO q := q + L 1 OD; (a < L 0 AND b > L 0 OR a ≥ L 0 AND b < L 0 | -q | q) FI; n) OP «÷×, ÷*, %*, MOD» = (L INT a, b) L INT: (L INT r = a - a ÷ b × b; r < 0 | r + ABS b | r);

[C-CPP-Std-5] Jones, Derek M. The New C Standard: An Economic and Cultural Commentary (PDF). Addison-Wesley. 2003 [2018-07-11]. ISBN 9780201709179. （原始内容 (PDF)存档于2018-07-11）（英语）.
C99 reflects almost universal processor behavior (as does the Fortran Standard). This definition truncates toward zero and the expression (-(a/b) == (-a)/b) && (-(a/b) == a/(-b)) is always true. It also means that the absolute value of the result does not depend on the signs of the operands; for example:

+10 / +3 == +3 +10 % +3 == +1 -10 / +3 == -3 -10 % +3 == -1
+10 / -3 == -3 +10 % -3 == +1 -10 / -3 == +3 -10 % -3 == -1
C90

When integers are divided and the division is inexact, if both operands are positive the result of the / operator is the largest integer less than the algebraic quotient and the result of the % operator is positive. If either operand is negative, whether the result of the / operator is the largest integer less than or equal to the algebraic quotient or the smallest integer greater than or equal to the algebraic quotient is implementation-defined, as is the sign of the result of the % operator.
If either operand is negative, the behavior may differ between C90 and C99, depending on the implementation-defined behavior of the C90 implementation.

#include <stdio.h> int main(void) { int x = -1, y = +3; if ((x%y > 0) || ((x+y)%y == x%y)) printf("This is a C90 translator behaving differently than C99\n"); }
Quoting from the C9X Revision Proposal, WG14/N613, that proposed this change:

The origin of this practice seems to have been a desire to map C’s division directly to the “natural” behavior of the target instruction set, whatever it may be, without requiring extra code overhead that might be necessary to check for special cases and enforce a particular behavior. However, the argument that Fortran programmers are unpleasantly surprised by this aspect of C and that there would be negligible impact on code efficiency was accepted by WG14, who agreed to require Fortran-like behavior in C99.

[6] Knuth, Donald. E. The Art of Computer Programming. Addison-Wesley. 1972 （英语）.
If $x$ is any real number, we write

$\left\lfloor x\right\rfloor$ = the greatest integer less than or equal to $x$ (the floor of $x$ );

$\left\lceil x\right\rceil$ = the least integer greater than or equal to $x$ (the ceiling of $x$ ).
……
The following formulas and examples are easily verified:
……

$\left\lceil x\right\rceil =\left\lfloor x\right\rfloor$ if and only if $x$ is an integer,

$\left\lceil x\right\rceil =\left\lfloor x\right\rfloor +1$ if and only if $x$ is not an integer;

$\left\lfloor -x\right\rfloor =-\left\lceil x\right\rceil$ ; $x-1<\left\lfloor x\right\rfloor \leq x\leq \left\lceil x\right\rceil <x+1$ .

……
If $x$ and $y$ are any real numbers, we define the following binary operation:

$x{\bmod {y}}=x-y\left\lfloor x/y\right\rfloor ,\ {\text{if }}y\neq 0$ ; $x{\bmod {0}}=x$ . (1)

From this definition we can see that, when $y\neq 0$ ,

$0\leq {\frac {x}{y}}-\left\lfloor {\frac {x}{y}}\right\rfloor ={\frac {x{\bmod {y}}}{y}}<1$ . (2)

Consequently
a) if $y>0$ , then $0\leq x{\bmod {y}}<y$ ;
b) if $y<0$ , then $0\geq x{\bmod {y}}>y$ ;
c) the quantity $x-(x{\bmod {y}})$ is an integral multiple of $y$ .
We call $x{\bmod {y}}$ the remainder when $x$ is divided by $y$ ; similarly, we call $\left\lfloor x/y\right\rfloor$ the quotient.
When $x$ and $y$ are integers, “ ${\text{mod}}$ ” is therefore a familiar operation:

$5{\bmod {3}}=2$ , $18{\bmod {3}}=0$ , $-2{\bmod {3}}=1$ . (3)

We have $x{\bmod {y}}=0$ if and only if $x$ is a multiple of $y$ , that is, if and only if $x$ is divisible by $y$ . The notation $y\backslash x$ , read “ $y$ divides $x$ ,” means that $y$ is a positive integer and $x{\bmod {y}}=0$ .
The “ ${\text{mod}}$ ” operation is useful also when $x$ and $y$ take arbitrary real values. For example, with trigonometric functions we can write

$\tan x=\tan(x{\bmod {\pi }})$ .

The quantity $x{\bmod {1}}$ is the fractional part of $x$ ; we have, by Eq. (1),

$x=\left\lfloor x\right\rfloor +(x{\bmod {1}})$ .

[7] Boute, Raymond T. The Euclidean definition of the functions div and mod. ACM Transactions on Programming Languages and Systems (ACM Press (New York, NY, USA)). April 1992, 14 (2): 127–144. doi:10.1145/128861.128862 （英语）.
Let us recall Euclid’s theorem.

THEOREM. For any real numbers $D$ and $d$ with $d\neq 0$ , there exists a unique pair of numbers $q,r$ satisfying the following conditions:

(a) $q\in \mathbb {Z}$ ;

(b) $D=d\cdot q+r$ ;

(c) $0\leq r<|d|$ .
For the particular case of integers, the conditions (a)-(c) correspond to part of the axioms for Euclidean rings [9].
With the notational conventions of the theorem, we define $D\operatorname {div} d=q$ and $D{\bmod {d}}=r$ . Clearly the correspondingly labeled conditions are satisfied.
Properties

${\begin{aligned}D\operatorname {div} \,(-d)&=-(D\operatorname {div} d)\\D{\bmod {\left(-d\right)}}&=D{\bmod {d}}\\(-D)\operatorname {div} d&=-(D\operatorname {div} d)-I\cdot J\\(-D){\bmod {d}}&=-(D{\bmod {d}})+d\cdot I\cdot J\end{aligned}}$
where $J=\operatorname {sign} d$ and $I$ is as defined earlier. ( $I={\text{if }}D/d\in \mathbb {Z} {\text{ then }}0{\text{ else }}1$ )

[Ada-8] 6.0 ^6.1 ISO/IEC 8652:2012 - Information technology — Programming languages — Ada. ISO, IEC. 2012. sec. 4.5.5 Multiplying Operators.

[C99-11] C99 specification (ISO/IEC 9899:TC2) (PDF). sec. 6.5.5 Multiplicative operators. 2005-05-06 [16 August 2018].

[12] 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

[14] ISO/IEC 9899:1990: Programming languages – C. ISO, IEC. 1990. sec. 7.5.6.4. The $fmod$ function returns the value $x - i * y$ , for some integer $i$ such that, if $y$ is nonzero, the result has the same sign as $x$ and magnitude less than the magnitude of $y$ .

[.NET-15] 10.0 ^10.1 dotnet-bot. Math.IEEERemainder(Double, Double) Method (System). Microsoft Learn. [2022-10-04] （美国英语）.

[16] ure.core - Clojure v1.10.3 API documentation. clojure.github.io. [2022-03-16].

[17] ure.core - Clojure v1.10.3 API documentation. clojure.github.io. [2022-03-16].

[isocobol-18] 13.0 ^13.1 ISO/IEC JTC 1/SC 22/WG 4. ISO/IEC 1989:2023 – Programming language COBOL. ISO. January 2023.

[CoffeeScript-19] CoffeeScript operators

[20] ISO/IEC JTC 1/SC 22. ISO/IEC 23271:2012 — Information technology — Common Language Infrastructure (CLI). ISO. February 2012. §§ III.3.55–56.

[21] Expressions - D Programming Language. dlang.org. [2021-06-01].

[22] rator % method - num class - dart:core library - Dart API. api.dart.dev. [2021-06-01].

[23] remainder method - num class - dart:core library - Dart API. api.dart.dev. [2021-06-01].

[24] Kernel — Elixir v1.11.3. hexdocs.pm. [2021-01-28].

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`div`	divide and truncate (i.e., value is not rounded)
`mod`	modulus: let `Remainder = A - (A div B) * B`; if `Remainder < 0` then `A mod B = Remainder + B` otherwise `A mod B = Remainder`