Complete Residue System

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.

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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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