Basic Example
The prototypical example of a congruence relation is congruence modulo on the set of integers. For a given positive integer, two integers and are called congruent modulo , written
if is divisible by (or equivalently if and have the same remainder when divided by ).
for example, and are congruent modulo ,
since is a multiple of 10, or equivalently since both and have a remainder of when divided by .
Congruence modulo (for a fixed ) is compatible with both addition and multiplication on the integers. That is, if
- and
then
- and
The corresponding addition and multiplication of equivalence classes is known as modular arithmetic. From the point of view of abstract algebra, congruence modulo is a congruence relation on the ring of integers, and arithmetic modulo occurs on the corresponding quotient ring.
Read more about this topic: Congruence Relation
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