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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.
Given the Euler's totient function φ(m), any set of φ(m) integers that are relatively prime to m and mutually incongruent under modulus m is called a reduced residue system modulo m. [5] The set {5, 15} from above, for example, is an instance of a reduced residue system modulo 4.
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 ...
Bulk modulus, a measure of compression resistance; Elastic modulus, a measure of stiffness; Shear modulus, a measure of elastic stiffness; Young's modulus, a specific elastic modulus; Modulo operation (a % b, mod(a, b), etc.), in both math and programming languages; results in remainder of a division; Casting modulus used in Chvorinov's rule.
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.
For any real number, the absolute value or modulus of is denoted by | |, with a vertical bar on each side of the quantity, and is defined as [8] | | = {,, < The absolute value of x {\displaystyle x} is thus always either a positive number or zero , but never negative .
Python's built-in pow() (exponentiation) function takes an optional third argument, the modulus .NET Framework 's BigInteger class has a ModPow() method to perform modular exponentiation Java 's java.math.BigInteger class has a modPow() method to perform modular exponentiation
If R is a ring, we can define the opposite ring R op, which has the same underlying set and the same addition operation, but the opposite multiplication: if ab = c in R, then ba = c in R op. Any left R-module M can then be seen to be a right module over R op, and any right module over R can be considered a left module over R op.