
Glossary: Prime Pages: Top 5000: 
Suppose a and m are any two integers with
m not zero. We say r is a residue of
a modulo m if a = r (mod
m). This is the same as m divides a 
r (see congruence), or a = r +
qm for some integer q. The division
algorithm tells us that there is a unique residue r
satisfying 0 < r < m, and
this remainder r is called the least
nonnegative residue of a modulo m.
A set of integers form a complete system of residues modulo m if every integer is congruent modulo m to exactly one integer in the set. So a complete system of residues includes exactly one element from each congruence class modulo m. For example, if m is positive, then {0, 1, 2, 3,..., m1}is a complete system of residues (called the least nonnegative residues modulo m). If m is positive and odd, then we sometimes use the system {  (m1)/2,  (m3)/2, ..., 1, 0, 1, ..., (m3)/2, (m1)/2}There are infinitely many complete residue systems for each modulus m.
Chris K. Caldwell © 19992017 (all rights reserved)
