Remainder of an integer

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modulo , residue of modulo

Any integer which is congruent to modulo (cf. Congruence). Let be the remainder of division of by some integer , ; then the residue of the number modulo will have the form , where is some integer. The residue corresponding to is equal to and is called the least non-negative residue of . The smallest (in absolute value) residue is called the absolutely smallest residue of . If , then ; if , then ; finally, if is even and , either or may be taken as .

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

A power residue of degree modulo , , is any integer , coprime with , for which the congruence

is solvable. If this congruence is not solvable, is called a power non-residue of degree modulo . In particular, if , the residues or non-residues are called quadratic; if , they are called cubic; if , 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