Euclidean division

In arithmetic, Euclidean division or division with remainder is the process of division of two integers, which produces a quotient and a remainder smaller than the divisor. Its main property is that the quotient and remainder exist and are unique, under some conditions. Because of this uniqueness, Euclidean division is often considered without referring to any method of computation, and without explicitly computing the quotient and the remainder. The methods of computation are called integer division algorithms, the best known being long division. In arithmetic, Euclidean division or division with remainder is the process of division of two integers, which produces a quotient and a remainder smaller than the divisor. Its main property is that the quotient and remainder exist and are unique, under some conditions. Because of this uniqueness, Euclidean division is often considered without referring to any method of computation, and without explicitly computing the quotient and the remainder. The methods of computation are called integer division algorithms, the best known being long division. Euclidean division, and algorithms to compute it, are fundamental for many questions concerning integers, such as the Euclidean algorithm for finding the greatest common divisor of two integers, and modular arithmetic, for which only remainders are considered. The operation consisting of computing only the remainder is called the modulo operation. Given two integers a and b, with b ≠ 0, there exist unique integers q and r such that