In computing, the modulo operation finds the remainder of division of one number by another.
Given two positive numbers, a and n, a modulo n is the remainder of the Euclidean division
of a by n. For instance, the expression "5 mod 2" would evaluate to 1 because 5 divided
by 2 leaves a quotient of 2 and a remainder of 1, while "9 mod 3" would evaluate to 0
because the division of 9 by 3 has a quotient of 3 and leaves a remainder of 0; there is
nothing to subtract from 9 after multiplying 3 times 3.
Although typically performed with a and n both being integers, many computing systems
allow other types of numeric operands. The range of numbers for an integer modulo of
n is 0 to n − 1. See modular arithmetic for an older and related convention applied
in number theory. When either a or n is negative, the naive
definition breaks down and programming languages differ in how these values are defined.
Remainder calculation for the modulo operation In mathematics the result of the modulo operation
is the remainder of the Euclidean division. 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 and/or the underlying hardware.
In nearly all computing systems, the quotient q and the remainder r satisfy
This means that, if the remainder is nonzero, there are two possible choices for the remainder,
one negative and the other positive, and there are also two possible choices for the quotient.
Usually, in number theory, the positive remainder is always chosen, but programming languages
choose depending on the language and the signs of a and n.[5] Pascal and Algol68 give a positive
remainder even for negative divisors, and 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), in other words it is the first integer in the direction of 0 from the exact
rational quotient, and the remainder by r=a − n q. Informally speaking the quotient
is "rounded towards zero", and the remainder therefore 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
Here the quotient is always rounded downwards and the remainder has the same sign as the
divisor. Raymond T. Boute introduces the Euclidean
definition, which is the one in which the remainder is always positive or 0, and is
therefore consistent with the division algorithm. This definition is marked as "Always positive"
in the table. Let q be the integer quotient of a and n, then:
Two corollaries are that
or, equivalently,
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 q=round(a/n),
q=ceil(a/n). IEEE 754 defines a remainder function where the quotient is a/n rounded
according to the round to nearest convention. Common pitfalls
When the result of a modulo operation has the sign of the dividend, it can sometimes
lead to surprising mistakes: For example, to test whether an integer is
odd, one might be inclined to test whether the remainder by 2 is equal to 1:
But in a language where modulo has the sign of the dividend, that is incorrect, because
when n is negative and odd, n % 2 returns −1, and the function returns false.
One correct alternative is to test that it is not 0:
Or, by understanding in the first place that for any odd number, the modulo remainder may
be either 1 or −1:
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
or equivalent, for environments lacking a mod() function
a -). Performance issues
Modulo operations might be implemented such that a division with a 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 &. 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.
Optimizing compilers may recognize expressions of the form expression % constant where constant
is a power of two and automatically implement them as expression &. This can allow the programmer
to write clearer code without compromising performance., because if the dividend is negative,
the modulo will be negative; however, expression & will always produce a positive result. So
special treatment has to be made when the dividend can be negative.)
Equivalencies Some modulo operations can be factored or
expanded similar to other mathematical operations. This may be useful in cryptography proofs,
such as the Diffie–Hellman key exchange. 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. Not defined otherwise.
Inverse Multiplication: See also
Modulo and modulo – many uses of the word "modulo", all of which grew out of Carl F.
Gauss's introduction of modular arithmetic in 1801.
Modular exponentiation Notes
^ Perl usually uses arithmetic modulo operator that is machine-independent. See the Perl
documentation for exceptions and examples. ^ Mathematically, these two choices are but
two of the infinite number of choices available for the inequality satisfied by a remainder.
^ Divisor must be positive, otherwise not defined.
^ As implemented in ACUCOBOL, Micro Focus COBOL, and possibly others.
References ^ open-std.org, section 6.5.5
^ ISO/IEC 14882:2003 : Programming languages -- C++. 5.6.4: ISO, IEC. 2003. . "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". ^ "Expressions". D Programming Language 2.0.
Digital Mars. Retrieved 29 July 2010. ^ a b r6rs.org
^ Knuth, Donald. E.. The Art of Computer Programming. Addison-Wesley.
^ Boute, Raymond T.. "The Euclidean definition of the functions div and mod". ACM Transactions
on Programming Languages and Systems) 14: 127–144. doi:10.1145/128861.128862.
^ Leijen, Daan. "Division and Modulus for Computer Scientists". Retrieved 2006-08-27.
^ Horvath, Adam. "Faster division and modulo operation - the power of two".