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.

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.)
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