814 THE THEORY OF NUMBERS 1800-1900 in the first section of Disquisitiones and used it systematically thereafter.The basic idea is simple.The number 27 is congruent to 3 modulo 4, 27 =3 modulo 4, because 27-3 is exactly divisible by 4.(The word modulo is often ab breviated to mod.)In general,if a,b,and m are integers a≡b modulo m if a-b is (exactly)divisible by m or if a and b have the same remainders on division by m.Then b is said to be a residue of a modulo m and a is a residuc of b modulo m.As Gauss shows,all the residues of a modulo m,for fixed a and lm,are given by a+km where k=0,±l,±2， Congruences with respect to the same modulus can be treated to some extent like cquations.Such congruences can be added,subtracted,and multiplied.One can also ask for the solution of congruences involving unknowns.Thus,what values of x satisfy 2x 25 modulo 12? This equation has no solutions because 2x is even and 2x-25 is odd.Hence 2x-25 cannot be a multiple of 12.The basic theorem on polynomial yedy en Ax+Bx-1+··+Mx+N=0 modulo p whose modulus is a prime number p which does not divide A cannot have more than n noncongruent roots. In the third section Gauss takes up residues of powers.Here he gives a tme f onmeot f tm a-1≡1 modulop. The theorem follows from his study of congruences of higher degree,namely, ≡a modulo m where a and m are relatively prime.This subject was continued by many men after Gauss. The fourth sctioof Disgreats qudratic reidues.Ifisa prime and a is not a multiple of and if there exists an x such that a mod,then a is a quadratic residue of;otherwise a is a quadratic non- residue of.After proving som subordinate theoremson quadratic residues Gauss gave the first rigorous proof of the law of quadratic reciprocity(Chap. 2.Hit.kl'4ad.de Berlin,24,1768,192,pub.1770=Ewr,2,655-726