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Equivalently, the elements of this group can be thought of as the congruence classes, also known as residues modulo n, that are coprime to n. Hence another name is the group of primitive residue classes modulo n. In the theory of rings, a branch of abstract algebra, it is described as the group of units of the ring of integers modulo n.
The multiplicative order of a number a modulo n is the order of a in the multiplicative group whose elements are the residues modulo n of the numbers coprime to n, and whose group operation is multiplication modulo n. This is the group of units of the ring Z n; it has φ(n) elements, φ being Euler's totient function, and is denoted as U(n) or ...
10) The multiplication and identity defined in 8) and the inversion defined in 9) turn the set of Dirichlet characters for a given modulus into a finite abelian group. The group of characters [ edit ]
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. 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. [1]
Congruence modulo m is a congruence relation, meaning that it is an equivalence relation that is compatible with the operations of addition, subtraction, and multiplication. Congruence modulo m is denoted a ≡ b (mod m). The parentheses mean that (mod m) applies to the entire equation, not just to the right-hand side (here, b).
If n is a positive integer, the integers from 1 to n − 1 that are coprime to n (or equivalently, the congruence classes coprime to n) form a group, with multiplication modulo n as the operation; it is denoted by × n, and is called the group of units modulo n, or the group of primitive classes modulo n.
The Sinclair ZX81 and its successors use the Lehmer RNG with parameters m = 2 16 + 1 = 65,537 (a Fermat prime F 4) and a = 75 (a primitive root modulo F 4). [7] [8] The CRAY random number generator RANF is a Lehmer RNG with the power-of-two modulus m = 2 48 and a = 44,485,709,377,909. [9]
gcd(r, n) = 1 for each r in R, R contains φ(n) elements, no two elements of R are congruent modulo n. [1] [2] Here φ denotes Euler's totient function. A reduced residue system modulo n can be formed from a complete residue system modulo n by removing all integers not relatively prime to n. For example, a complete residue system modulo 12 is ...