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In modular arithmetic, the integers coprime (relatively prime) to n from the set of n non-negative integers form a group under multiplication modulo n, called the multiplicative group of integers modulo n. 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.
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 Zn; it has φ (n) elements, φ being Euler's totient function, and is denoted as U (n) or ...
n, and is called the group of units modulo n, or the group of primitive classes modulo n. As explained in the article multiplicative group of integers modulo n, this multiplicative group (× n) is cyclic if and only if n is equal to 2, 4, p k, or 2 p k where p k is a power of an odd prime number.
Adding 4 hours to 9 o'clock gives 1 o'clock, since 13 is congruent to 1 modulo 12. In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones ...
The group scheme of n-th roots of unity is by definition the kernel of the n-power map on the multiplicative group GL(1), considered as a group scheme.That is, for any integer n > 1 we can consider the morphism on the multiplicative group that takes n-th powers, and take an appropriate fiber product of schemes, with the morphism e that serves as the identity.
Euler's totient function is a multiplicative function, meaning that if two numbers m and n are relatively prime, then φ(mn) = φ(m)φ(n). [4][5] This function gives the order of the multiplicative group of integers modulo n (the group of units of the ring ). [6] It is also used for defining the RSA encryption system.
Important subgroups of the modular group Γ, called congruence subgroups, are given by imposing congruence relations on the associated matrices. There is a natural homomorphism SL(2, Z) → SL(2, Z/NZ) given by reducing the entries modulo N. This induces a homomorphism on the modular group PSL(2, Z) → PSL(2, Z/NZ).
In number theory, a branch of mathematics, the Carmichael function λ(n) of a positive integer n is the smallest positive integer m such that. holds for every integer a coprime to n. In algebraic terms, λ(n) is the exponent of the multiplicative group of integers modulo n. As this is a finite abelian group, there must exist an element whose ...