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In computing and mathematics, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, the latter being called the modulus of the operation So, it must be moved to modulo operation (programming). 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

Implements the mathematical modulo operator Modulo operation is about computer implementations of the modulo operation defined in modular arithmetics The returned result is always of the same sign as the modulus or nul, and its absolute value is lower than the absolute value of the modulus

However, this template returns 0 if the modulus is nul (this template should never return a division by zero error).

The modulo operation, as implemented in many programming languages and calculators, is an application of modular arithmetic that is often used in this context The logical operator xor sums 2 bits, modulo 2. 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 modulo operation is the operation that produces such a remainder when given a dividend and divisor Alternatively, a remainder is also what is left after subtracting one number from another, although this is more precisely called the difference. Definition 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.

Applications mod operator for an integer x and a positive integer y, the modulo operation, denoted by x mod y, gives the value of the remainder when x is divided by y

This definition can be extended to real x and y, y ≠ 0, by the formula x mod y = x − y ⌊ x y ⌋ In modular arithmetic computation, montgomery modular multiplication, more commonly referred to as montgomery multiplication, is a method for performing fast modular multiplication It was introduced in 1985 by the american mathematician peter l [1][2] montgomery modular multiplication relies on a special representation of numbers called montgomery form

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