Difference between revisions of "Modular arithmetic"
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| − | '''Modular arithmetic''' a special type of arithmetic that involves only [[integers]]. If two integers <math>{a},{b}</math> leave the same remainder when they are divided by some positive integer <math>{m}</math>, we say that <math>{a}</math> and <math>b</math> are congruent [[modulo]] <math>{m}</math> or <math>a\equiv b \pmod {m}</math>. | + | '''Modular arithmetic''' a special type of arithmetic that involves only [[integers]]. If two integers <math>{a},{b}</math> leave the same remainder when they are divided by some positive integer <math>{m}</math>, we say that <math>{a}</math> and <math>b</math> are congruent [[modulo]] <math>{m}</math> or <math>a\equiv b \pmod {m}</math>. Sometimes we refer to the integers modulo n. This is symbolically represented by <math>\mathbb{Z}_n</math>. |
Revision as of 11:44, 23 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 
. Sometimes we refer to the integers modulo n. This is symbolically represented by 
.
Contents
[hide]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.
.
.
.
, 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.
