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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 It might be outdated or ideologically biased.



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.



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.


Vinogradov, I. M. Osnovy teorii chisel, 8th ed. Moscow, 1972.
Hasse, H. Lektsii po teorii chisel. Moscow, 1953. (Translated from German.)
The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
The hypotheses H1 and H1a predicted that sex and gender role incongruence were significantly related to diagnosis.
More passive children will absorb what is expected of them and fall in line, without communicating much about their feelings of gender incongruence.
In official comments submitted to APA, Winters (2010) praised the less stigmatizing name of the condition, the emphasis on "gender incongruence," and the requirement of gender dysphoria for diagnosis.
This pattern of incongruence suggests that the public is quite pessimistic about the quality of citizenship education in American high schools, a pattern in line with existing public opinion research on how citizens perceive the quality of "public schools" in general.
(1) The best available evidence points to transgender children being born that way, but society (and science to a lesser extent) tend to view the incongruence as a 'disorder', attributing it to more esoteric reasons, like divorce, parental neglect, parents wishing they had given birth to the other sex, using fertility drugs to conceive, encouraging children to play sports too often or not enough or a range of other parental thoughts, behaviours or influences.
This study examined the relationship between Holland type (Realistic, Investigative, Artistic, Social, Enterprising, and Conventional; Holland, 1959, 1997) congruence and incongruence (i.e., lack of fit between an occupation's 3-letter Holland code and a person's lowest 3 Holland interest types) and tested whether incongruence predicts unique variance in satisfaction beyond congruence.
The empirical evidence suggests both that self-selection into a discipline is one mechanism by which strong person-organization congruence is achieved (Sidanius, Pratto, Sinclair, & van Laar, 1996; Sidanius et al., 2003) and that at least some percentage of individuals who initially choose a field of study experience cultural incongruence with the values and predominant attitudes of the discipline into which they are entering.
This incongruence inhibits the gateways from passing the necessary data between platforms, adding downgrade fees to the transaction and costing Energy marketers thousands of dollars every month.
Transgender identity can be established in the clinical setting on the basis of history alone; patients should have persistent gender incongruence as well as the capacity to make medical decisions.