Euclidean division

In arithmetic, the Euclidean division is the process of division of two integers, which produces a quotient and a remainder. There is a theorem stating that the quotient and remainder exist, are unique, and have certain properties. Integer division algorithms compute the quotient and remainder given two integers, the most well-known such algorithm 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 in computing only the remainder is called the modulo operation.

Intuitive example

Suppose that a pie has 9 slices and they are to be divided evenly among 4 people. Using Euclidean division, 9 divided by 4 is 2 with remainder 1. In other words, each person receives 2 slices of pie, and there is 1 slice left over.

This can be confirmed using multiplication, the inverse of division: if each of the 4 people received 2 slices, then 4 × 2 = 8 slices were given out in all. Adding the 1 slice remaining, the result is 9 slices. In summary: 9 = 4 × 2 + 1.