# Congruence

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## congruence

[kən′grü·əns]
(mathematics)
The property of geometric figures that can be made to coincide by a rigid transformation. Also known as superposability.
The property of two integers having the same remainder on division by another integer.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## Congruence

a term used in geometry to denote the equality of segments, angles, triangles, and other figures and solids in elementary geometry. The concept of congruence may be taken as one of the undefined terms of elementary geometry. Its properties may, in this case, be characterized by appropriate axioms, which are called the axioms of congruence. If, instead, we take motion as an undefined term (seeMOTION), then the concept of congruence can be given a direct definition: two figures are congruent if one of them can be transformed into the other by means of motion.

## Congruence

the relation between two integers a and b that consists in the difference a – b between the numbers being divisible by some given number m, which is called the modulus of the congruence. The numbers a and b are said to be congruent modulo m; this statement is usually written a ≡ b (mod tri). Since, for example, 2 – 8 is divisible by 3, we have 2 ≡ 8 (mod 3).

Congruences are similar in many of their properties to equalities. For example, a term on one side of a congruence can be transposed to the other side, where it will have the opposite sign—that is, it follows from a + b ≡ c (mod m) that a ≡ c – b (mod m). Congruences with the same modulus can be added, subtracted, and multiplied—that is, if a ≡ b (mod m) and c ≡ d (mod m), then a + c ≡ b + d (mod m), a – c ≡ b – d (mod m), and ac ≡ bd (mod m). Furthermore, both sides of a congruence can be multiplied by the same integer. Both sides of a congruence can be divided by a common divisor if the divisor and the modulus are relatively prime. If, however, the number d is the greatest common divisor of the modulus m and of a number by which both sides of the congruence are divided, then a congruence with respect to the modulus mid is obtained when the division is performed.

Methods of solving various congruences are dealt with in number theory. The solution of a congruence involves finding an integer that satisfies the congruence. If the number x is a solution of some congruence modulo m, then any number of the form x + km, where k is an integer, is also a solution of the congruence. A set of numbers of the form x+ km, where k =...,–1,0,1, . . . , is called a residue class modulo m. Solutions of a congruence modulo m that belong to the same residue class are not regarded as distinct. Thus, the number of solutions of a congruence modulo m is understood as the number of solutions that belong to different residue classes. A first-degree congruence in one unknown can always be reduced to the form ax ≡ b (mod m). Such a congruence has no solution if b is not divisible by the greatest common divisor d of a and m; the congruence has d solutions if b is divisible by d.

The theory of quadratic residues and power residues modulo m is concerned with congruences of the form x2 ≡ a (mod m) and xna (mod m), respectively. The concept of the congruence of integers can be extended. Thus, we can speak of the congruence of two elements of a ring with respect to an ideal.

### REFERENCES

Vinogradov, I. M. Osnovy teorii chisel, 8th ed. Moscow, 1972.
Hasse, H. Lektsii po teorii chisel. Moscow, 1953. (Translated from German.)
References in periodicals archive ?
These studies have overwhelmingly focused on the ideological congruence of governments: how close cabinets are to citizens on the left-right dimension.
H3: Congruence on prototypes will affect LMX positively.
Self-image congruence refers to the cognitive match between a consumer's self-concept and the stereotyped user image of a given brand, product, or retail store (Kressmann et al., 2006).
A customer can identify with a door-to-door salesperson based on appearance and self-image congruence. Consumers from a more affluent socioeconomic status see themselves as more congruent with a professionally dressed salesperson.
The first congruence implies d [equivalent to] [I.sub.g] (mod 4), from [a.sup.t]d - [b.sup.t]c = [I.sub.g], and we get a [equivalent to] [I.sub.g] (mod 4).
For any fixed positive integer h, we prove this polynomial congruence by complete induction for positive integer n.
Since the United States has been a strong ally of South Korea and still plays an important role in East Asian affairs, we expect that US interests strongly influence vote congruence between South Korea and China.
In this case, the results were quite similar to those obtained when both biases were removed, and the congruence indexes were similar.
In order to provide an additional discussion of the reasons behind the adverse effects of the proposed congruence, this book could refer to the literature focusing on the effect of power-sharing arrangements on the shortterm conflict-managing and the long-term state-building.
Among situational moderators the medical evidence, relevant to the patient's condition and available to the caregiver when making symptom judgments, was proposed as a possible bias source for PSRP-NEP congruence. Medical evidence includes all medical findings: symptoms, diagnosis, and treatments.
 investigated the function [C.sub.N] (n) for N = 3 and 4 and proved some congruence properties modulo 5.
Test-retest reliability is the psychometric property most similar to thematic congruence, except the focus of the former is on identical replication whereas the latter is a broader concept encompassing similarity or compatibility of meaning (Maille & Fleck, 2011).

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