Editing Congruence relation (section)

=== Basic example ===
{{About|the ''(mod'' n'')'' notation|the binary ''mod'' operation|modulo operation|section=yes}}

The prototypical example of a congruence relation is [[Modular arithmetic#Congruence|congruence modulo]] <math>n</math> on the set of [[integer]]s.  For a given positive integer <math>n</math>, two integers <math>a</math> and <math>b</math> are called '''congruent modulo <math>n</math>''', written
: <math>a \equiv b \pmod{n}</math>
if <math>a - b</math> is [[divisible]] by <math>n</math> (or equivalently if <math>a</math> and <math>b</math> have the same [[remainder]] when divided by <math>n</math>).

For example, <math>37</math> and <math>57</math> are congruent modulo <math>10</math>,
: <math>37 \equiv 57 \pmod{10}</math>
since <math>37 - 57 = -20</math> is a multiple of 10, or equivalently since both <math>37</math> and <math>57</math> have a remainder of <math>7</math> when divided by <math>10</math>.

Congruence modulo <math>n</math> (for a fixed <math>n</math>) is compatible with both [[addition]] and [[multiplication]] on the integers.  That is,

if
: <math>a_1 \equiv a_2 \pmod{n} </math> and <math> b_1 \equiv b_2 \pmod{n}</math> 
then
: <math>a_1 + b_1 \equiv a_2 + b_2 \pmod{n} </math>  and  <math> a_1 b_1 \equiv a_2b_2 \pmod{n}</math>

The corresponding addition and multiplication of equivalence classes is known as [[modular arithmetic]].  From the point of view of abstract algebra, congruence modulo <math>n</math> is a congruence relation on the [[ring (mathematics)|ring]] of integers, and arithmetic modulo <math>n</math> occurs on the corresponding [[quotient ring]].