Complete Residue System

The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Complete Residue System

 

Any set of integers containing one representative from each number class modulo m (two integers a and b belong to the same class modulo m if a – b is divisible by m) is said to be a complete residue system modulo m. The most frequently used complete residue systems are the system of smallest positive residues 0, 1, 2, …, m – 1 and the system of absolutely smallest residues: – (m – 1)/2, …, -1, 0, 1, …, (m - 1)/2 for odd m and -m/2, – 1, 0, 1, …, m/2 – 1 for even m. Any m numbers that belong to different residue classes modulo m form a complete residue system modulo m.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
Now taking s = a/q in (10), and summation for all 1 [less than or equal to] a [less than or equal to] q - 1 and noting the definition and properties of complete residue system mod q (see [16]) we have
Since (h, q) = 1, if a pass through a complete residue system mod q, then ha also pass through a complete residue system mod q.
If S is a complete residue system (mod n), then we say that the set S', consisting of all elements of S except 0 (mod n), is a complete system of nonzero residues (mod n).

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