Modular arithmetic
Revision as of 11:44, 23 June 2006 by IntrepidMath (talk | contribs)
Modular arithmetic a special type of arithmetic that involves only integers. If two integers leave the same remainder when they are divided by some positive integer
, we say that
and
are congruent modulo
or
. Sometimes we refer to the integers modulo n. This is symbolically represented by
.
Contents
Introductory
Useful Facts
Consider four integers and a positive integer
such that
and
. In modular arithmetic, the following identities hold:
- Addition:
.
- Substraction:
.
- Multiplication:
.
- Division:
, where
is a positive integer that divides
and
.
- Exponentiation:
where
is a positive integer.
Examples
Applications
Modular arithmetic is an extremely useful tool in mathematics competitions. It enables us to easily solve Linear diophantine equations, and it often helps with other Diophantine equations as well.