Remainder of an integer

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$a$ modulo $m$, residue of $a$ modulo $m$

Any integer $b$ which is congruent to $a$ modulo $m$ (cf. Congruence). Let $r$ be the remainder of division of $a$ by some integer $m>0$, $0\leq r\leq m-1$; then the residue $b$ of the number $a$ modulo $m$ will have the form $b=mq+r$, where $q$ is some integer. The residue corresponding to $q=0$ is equal to $r$ and is called the least non-negative residue of $a$. The smallest (in absolute value) residue $\rho$ is called the absolutely smallest residue of $a$. If $r<m/2$, then $\rho=r$; if $r>m/2$, then $\rho=r-m$; finally, if $m$ is even and $r=m/2$, either $m/2$ or $-m/2$ may be taken as $\rho$.

A system consisting of $m$ integers each one of which is the residue of one and only one of the numbers $0,\ldots,m-1$ is called a complete system of residues modulo $m$. The smallest non-negative residues $0,\ldots,m-1$ or the absolutely smallest residues are the complete systems of residues which are most frequently used.

A power residue of degree $n$ modulo $m$, $n\geq2$, is any integer $a$, coprime with $m$, for which the congruence

$$x^n\equiv a\pmod m$$

is solvable. If this congruence is not solvable, $a$ is called a power non-residue of degree $n$ modulo $m$. In particular, if $n=2$, the residues or non-residues are called quadratic; if $n=3$, they are called cubic; if $n=4$, they are called biquadratic (see also Power residue).


[1] I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian)



[a1] G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapt. XIII
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Remainder of an integer. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article