“congruence”的概念、定义、习题解题方法及翻译-数学辞典

单词

congruence

释义

congruence

(modulo n) For each integer n ≥ 2, the relation of congruence between integers is defined as a is congruent to b modulo n if a−b is a multiple of n. This is written a ≡ b (mod n). Then a ≡ b (mod n) if and only if a and b have the same remainder (or residue) upon division by n. For example, 19 is congruent to 7 modulo 3. The following properties hold, if a ≡ b (mod n) and c ≡ d (mod n):

(i) a + c ≡ b + d (mod n),

(ii) a − c ≡ b − d (mod n),

(iii) ac ≡ bd (mod n).

That is, addition, subtraction, and multiplication are well defined; division in general is not.The integer n is the modulus of the congruence. Congruence modulo n is an equivalence relation, with the n equivalence classes usually represented by 0,1,2,…,n − 1. The set of these residue classes is denoted ℤn and form a commutative ring for all n and a field if n is prime. See modulo n arithmetic.

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