Complete residue system

Japanese: 剰余系 - じょうよけい（英語表記）complete residue system

For a natural number m , if the remainders when two integers a and b are divided by m are equal, then a and b are said to be congruent modulo m . If all integers are classified according to this relationship, they are divided into m classes. These classes are called coset classes, and a set of integers obtained by taking one element each from each class is called a coset system. For example, if m = 6, then 0, 1, 2, 3, 4, and 5 are one set of coset systems. 6, -5, 8, 9, -2, and -1 are also coset systems. A class consisting of integers relatively prime to m is called an irreducible coset class, and the set of integers obtained by selecting one element each from each class is called an irreducible coset system.

Source: Heibonsha World Encyclopedia, 2nd Edition Information

Japanese:

自然数mに対して，二つの整数aとbのmで割った余りが等しいとき，aとbはmを法として合同であるという。この関係で整数全体をクラス分けすると，m個の類(クラス)に分かれる。これらの類を剰余類といい，この各類から一つずつ元を取り出した組を剰余系という。例えばm＝6とすれば，0,1,2,3,4,5は1組の剰余系である。また，6，－5,8,9，－2，－1も剰余系である。mと互いに素な整数からなる類を既約剰余類といい，その各類から一つずつ元を選び出して得られた整数の組を既約剰余系という。

出典　株式会社平凡社世界大百科事典第２版について　情報

<<: Remainder theorem

>>: Surplus - joyokin (English spelling) surplus