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   CONGRUENCE, RESIDUE CLASSES OF INTEGERS MODULO N

Congruence. Let n be a positive integer. We say two integers are congruent “modulo n" if they differ by an integral multiple of the integer n. For example, if n = 5 we can say that 3 is congruent to 23 modulo 5 (and write it as 3 23 modulo 5) since the integers 3 and 23 differ by 4x5 = 20. The statement a b (mod n) is equivalent to the statement “a - b is divisible by n” or the statement “there is an integer k for which a - b = kn”. The integer n is called the modulus of the congruence.

An alternate definition: a b (mod n) if and only if a and b have the same remainder when divided by n.

The modulus arithmetic concept occurs in everyday life in telling time. Clocks go up to12 and then start over, thus giving time modulus 12.

The congruence relation a b (mod n) creates a set of equivalence classes on the set of integers in which two integers are in the same class if they are congruent modulus n, i.e. if they leave the same remainder when divided by n.

Syn. Modulus, modulo, mod

Residue classes of integers mod n. The congruence relation a b (mod n) on the set of integers I separates the integers into n equivalence classes,

                                           [0]_n, [1]_n, [2]_n, ... ,[n-1]_n,

called residue classes modulo n. Each equivalence class [r]_n consists of all integers congruent to r where r is one of the integers 0, 1, 2, ... ,n-1. These n integers 0, 1, 2, ... ,n-1 are called the class representatives. Thus equivalence class [3]_n consists of all integers congruent to 3 mod n where the integer 3 is the class representative.

Example. The residue classes of integers mod 4 are:

            [0]₄ = { ... , -16, -12, -8, -4, 0, 4, 8, 12, 16, ... }

            [1]₄ = { ... , -15, -11, -7, -3, 1, 5, 9, 13, 17, ... }

            [2]₄ = { ... , -14, -10, -6, -2, 2, 6, 10,14, 18, ... }

            [3]₄ = { ... , -13, -9, -5, -1, 3, 7, 11, 15, 19, ... }

I/(n), the set of all residue classes mod n. We denote the set of all residue classes modulo n by I/(n). For example,

            I/(4) = { [0]₄, [1]₄, [2]_4, [3]₄ }

and

          I/(n) = { [0]_n,, [1]_n,, [2]_n,, ... ,[n-1]_n, }

Note that I/(n) consists of a set of sets.

Modular arithmetic (or arithmetic modulo n). A modular arithmetic or arithmetic modulo n is obtained by using only the class representatives 0, 1, 2, ... ,n-1 and defining addition and multiplication by letting the sum a + b and the product ab be the remainder after division by n of the ordinary sum and product of a and b. E.g. if n = 7, then 2 + 6 1, 3∙6 4, and the multiplicative inverse of 2 is 4, since 2∙4 1. Multiplicative inverses need not exist, however. For example, if n = 15, then 3 has no multiplicative inverse, since a multiplicative inverse a would need to meet the condition that 3∙a - k∙15 = 1 or, equivalently, a - 5k = 1/3 for some set of integers a and k. But there are no set of integers a and k that will meet this condition.

Arithmetic modulo n is a commutative ring with unit element. If n is a prime, then arithmetic modulo n is a field.

References

  Saunders, MacLane. A Survey of Modern Algebra. p. 23 - 29

  Ayres. Modern Algebra. p.53

  James & James. Mathematics Dictionary. “Congruence”