residue (another Prime Pages' Glossary entries) Glossary: Prime Pages: Top 5000: GIMPS has discovered a new largest known prime number: 282589933-1 (24,862,048 digits)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,..., m-1} 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 { - (m-1)/2, - (m-3)/2, ..., -1, 0, 1, ..., (m-3)/2, (m-1)/2} There are infinitely many complete residue systems for each modulus m. Chris K. Caldwell © 1999-2020 (all rights reserved)