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Inverse multiplication: mod n = a mod n.

Division (definition): a / b mod n = mod n, when the right hand side is defined (that is when b and n are coprime), and undefined otherwise.

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: mod n = 1.

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.

n x mod n = 0 for all positive integer values of x.

Some of these properties require that a and n are integers. This may be useful in cryptography proofs, such as the Diffie–Hellman key exchange. Some modulo operations can be factored or expanded similarly to other mathematical operations. See also: Modular arithmetic § Properties Optimizations for general constant-modulus operations also exist by calculating the division first using the constant-divisor optimization. For these languages, the equivalence x % 2 n = x < 0 ? x | ~(2 n - 1) : x & (2 n - 1) has to be used instead, expressed using bitwise OR, NOT and AND operations.

This is because, if the dividend is negative, the modulo will be negative, whereas expression & (constant-1) will always be positive. 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. Ĭompiler 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. In devices and software that implement bitwise operations more efficiently than modulo, these alternative forms can result in faster calculations. 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): For special cases, on some hardware, faster alternatives exist. Modulo operations might be implemented such that a division with a remainder is calculated each time. Many implementations use truncated division, for which the quotient is defined by Quotient ( q) and remainder ( r) as functions of dividend ( a), using truncated division.In nearly all computing systems, the quotient q and the remainder r of a divided by n satisfy the following conditions: 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 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). When exactly one of a or n is negative, the basic definition breaks down, and programming languages differ in how these values are defined. The range of values for an integer modulo operation of n is 0 to n − 1 ( a mod 1 is always 0 a mod 0 is undefined, being a division by zero). įor example, the expression '5 mod 2' evaluates to 1, because 5 divided by 2 has a quotient 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.Īlthough typically performed with a and n both being integers, many computing systems now allow other types of numeric operands.

Given two positive numbers a and n, a modulo n (often abbreviated as a mod n) is the remainder of the Euclidean division of a by n, where a is the dividend and n is the divisor.

In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the modulus of the operation).

Matlab mod