Difference between revisions of "Modular arithmetic"
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=== Topics === | === Topics === | ||
* [[Fermat's Little Theorem]] | * [[Fermat's Little Theorem]] | ||
| − | * [[Euler's Theorem]] | + | * [[Euler's Totient Theorem]] |
* [[Phi function]] | * [[Phi function]] | ||
| − | |||
=== See also === | === See also === | ||
Revision as of 13:47, 19 June 2006
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 
.
Contents
Introductory
Operations
Consider four integers 
and a positive integer such that 
and 
. In modular arithmetic, the following operations are allowed:
- Addition:
. - Substraction:
. - Multiplication:
. - Division:
, where 
is a positive integer that divides 
and 
. - Exponentiation:
where is a positive integer.
.
.
.
, where 
is a positive integer that divides 
where 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.
