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congruence


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(modulo n)

For each positive integer n, the relation of congruence between integers is defined as follows: a is congruent to b modulo n if ab is a multiple of n. This is written ab (mod n). The integer n is the modulus of the congruence. Then ab (mod n) if and only if a and b have the same remainder upon division by n. For example, 19 is congruent to 7 modulo 3. The following properties hold, if ab (mod n) and cd (mod n):(i)a+cb+d (mod n),(ii)acbd (mod n),(iii)acbd (mod n).It can be shown that congruence modulo n is an equivalence relation and so defines a partition of the set of integers, where two integers are in the same class if and only if they are congruent modulo n. These classes are the residue (or congruence) classes modulo n.

(i)a+cb+d (mod n),

(ii)acbd (mod n),

(iii)acbd (mod n).

Subjects: Mathematics.


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