Glossary:
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In our entry on congruences we note that if m is
not zero and a, b and c are any
integers, then we have the following:
- The reflexive property: If a is any
integer, a
 a (mod m),
- The symmetric property: If a b
(mod m), then b a (mod m),
- The transitive property: If a
b (mod m) and b c (mod
m), then a c (mod m).
These three properties
are just what we need to show that the integers are
divided into exactly m congruence classes
containing integers mutually congruent modulo m.
(Technically, we say congruence is an
equivalence relation.) For example, modulo
five we have the 5 classes
- ... -10 -5 0 5 10 15 ... (mod 5)
- ... -9 -4 1 6 11 16 ... (mod 5)
- ... -8 -3 2 7 12 17 ... (mod 5)
- ... -7 -2 3 8 13 18 ... (mod 5)
- ... -6 -1 4 9 14 19 ... (mod 5)
Modulo two there are the two classes we call the even and odd integers:
- ... -4 -2 0 2 4 6 ... (mod 2)
- ... -3 -1 1 3 5 7 ... (mod 2)
Sometimes we denote these classes as 0 mod 2, and
1 mod 2 respectively.
See Also: Residue
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