Ad
related to: congruence and modulus formula calculator
Search results
Results From The WOW.Com Content Network
The congruence relation is an equivalence relation. The equivalence class modulo m of an integer a is the set of all integers of the form a + k m, where k is any integer. It is called the congruence class or residue class of a modulo m, and may be denoted as (a mod m), or as a or [a] when the modulus m is known from the context.
Integers in the same congruence class a ≡ b (mod n) satisfy gcd(a, n) = gcd(b, n); hence one is coprime to n if and only if the other is. Thus the notion of congruence classes modulo n that are coprime to n is well-defined. Since gcd(a, n) = 1 and gcd(b, n) = 1 implies gcd(ab, n) = 1, the set of classes coprime to n is closed under ...
The congruence relation, modulo m, partitions the set of integers into m congruence classes. Operations of addition and multiplication can be defined on these m objects in the following way: To either add or multiply two congruence classes, first pick a representative (in any way) from each class, then perform the usual operation for integers on the two representatives and finally take the ...
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]
Modulo is a mathematical jargon that was introduced into mathematics in the book Disquisitiones Arithmeticae by Carl Friedrich Gauss in 1801. [3] Given the integers a, b and n, the expression "a ≡ b (mod n)", pronounced "a is congruent to b modulo n", means that a − b is an integer multiple of n, or equivalently, a and b both share the same remainder when divided by n.
The general formula is X k + 1 = a ⋅ X k mod m , {\displaystyle X_{k+1}=a\cdot X_{k}{\bmod {m}},} where the modulus m is a prime number or a power of a prime number , the multiplier a is an element of high multiplicative order modulo m (e.g., a primitive root modulo n ), and the seed X 0 is coprime to m .
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.
For computational purposes it is also necessary that division and reduction modulo R are inexpensive, and the modulus is not useful for modular multiplication unless R > N. The Montgomery form of the residue class a with respect to R is aR mod N , that is, it is the representative of the residue class aR .