# 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*.