Modulo operation

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, i.e. 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 or the underlying hardware.

In nearly all computing systems, the quotient q and the remainder r of a divided by n satisfy

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

(1)

However, this still leaves a sign ambiguity if the remainder is nonzero: two possible choices for the remainder occur, one negative and the other positive, and two possible choices for the quotient occur. Usually, in number theory, the positive remainder is always chosen, but programming languages choose depending on the language and the signs of a or n. Standard Pascal and ALGOL 68 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 for details. a modulo 0 is undefined in most systems, although some do define it as a.

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.

— Daan Leijen, Division and Modulus for Computer Scientists[3]

However, Boute concentrates on the properties of the modulo operation itself and does not rate the fact that the truncated division shows the symmetry (-a) div n = -(a div n) and a div (-n) = -(a div n), which is similar to the ordinary division. As neither floor division nor Euclidean division offer this symmetry, Boute's judgement is at least incomplete.

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(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;
}

Another is to use the fact that, for any odd number, the remainder may be either 1 or −1:

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

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

or equivalent, for environments lacking a mod() function ('int' inherently produces the truncated value of a/n)

a - (n * int(a/n))

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:

x % 2n == x & (2n - 1)

Examples (assuming x is a positive integer):

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.[4]

Optimizing compilers 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.

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.

• 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). Undefined otherwise.
• Inverse multiplication: [(ab mod n)(b⁻¹ mod n)] mod n = a mod n.

• Modulo (disambiguation) and modulo (jargon) – many uses of the word modulo, all of which grew out of Carl F. Gauss's introduction of modular arithmetic in 1801.
• Modular exponentiation
• Turn (unit)

• ^ Perl usually uses arithmetic modulo operator that is machine-independent. For examples and exceptions, see the Perl documentation on multiplicative operators.[13]
• ^ 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 undefined.
• ^ As implemented in ACUCOBOL, Micro Focus COBOL, and possible others.
• ^ ^ Argument order reverses, i.e., α|ω computes ${\displaystyle \omega {\bmod {\alpha }}}$, the remainder when dividing ω by α.
• ^ As discussed by Boute, ISO Pascal's definitions of div and mod do not obey the Division Identity, and are thus fundamentally broken.

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