Explore: Residue Classes

belongs to that class (since this is the proper remainder which results from division). Any two members of different residue classes modulo m are incongruent

Gaussian integers into equivalence classes, called here congruence classes or residue classes. The set of the residue classes is usually denoted Z[i]/z0Z[i]

algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group

half of all residue classes of numbers are removed, as opposed to small sieves such as the Selberg sieve wherein only a few residue classes are removed

product, and is equal to 1. Commutative rings have a product operation. Residue classes in the rings Z / N Z {\displaystyle \mathbb {Z} /N\mathbb {Z} } can

then a is in one of these residue classes, and its powers a, a2, ... , ak modulo n form a subgroup of the group of residue classes, with ak ≡ 1 (mod n). Lagrange's

residue class unless they would violate the divisibility condition). The residue class corresponding to a is denoted a. Equality of residue classes is

from a 1748 paper by Leonhard Euler. The proof uses the fact that the residue classes modulo a prime number are a field. See the article prime field for

combinatorial mathematics from Penn in 1993. His thesis was On the Residue Classes of Combinatorial Families of Numbers. Shortly thereafter, he became

{\displaystyle p\geq 3} . Since the residue classes modulo p {\displaystyle p} form a field, every non-zero residue a {\displaystyle a} has a unique multiplicative