residue

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.