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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. In general, if the number of slices is denoted ''a'' and the number of people is ''b'', one can divide the pie evenly among the people such that each person receives ''q'' slices (the quotient) and some number of slices ''r'' < ''b'' are left over (the remainder). Regardless, the equation ''a'' = ''bq'' + ''r'' holds. If 9 slices were divided among 3 people instead of 4, each would receive 3 and no slices would be left over. In this case the remainder is zero, and it is said that 3 ''evenly divides'' 9, or that 3 ''divides'' 9. Euclidean division can also be extended to negative integers using the same formula; for example −9 = 4 × (−3) + 3, so −9 divided by 4 is −3 with remainder 3. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Euclidean division」の詳細全文を読む スポンサード リンク
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