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n. 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.
Quotient group. A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored out"). For example, the cyclic group of addition modulo n can be obtained from the group of integers ...
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 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 ...
For every positive integer n, the set of integers modulo n, again with the operation of addition, forms a finite cyclic group, denoted Z/nZ. [1] A modular integer i is a generator of this group if i is relatively prime to n , because these elements can generate all other elements of the group through integer addition.
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.
Other important quotients are the (2, 3, n) triangle groups, which correspond geometrically to descending to a cylinder, quotienting the x coordinate modulo n, as T n = (z ↦ z + n). (2, 3, 5) is the group of icosahedral symmetry, and the (2, 3, 7) triangle group (and associated tiling) is the cover for all Hurwitz surfaces.
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]