Difference between revisions of "Modular arithmetic"
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− | '''Modular arithmetic''' a special type of arithmetic that involves only [[integers]]. | + | '''Modular arithmetic''' is a special type of arithmetic that involves only [[integers]]. Given integers <math>a</math>, <math>b</math>, and <math>n</math>, with <math>n > 0</math>, we say that <math>a</math> is ''congruent to'' <math>b</math> ''modulo'' <math>n</math>, or <math>a \equiv b</math> (mod <math>n</math>), if the difference <math>{a - b}</math> is divisible by <math>n</math>. |
Revision as of 20:47, 23 June 2006
Modular arithmetic is a special type of arithmetic that involves only integers. Given integers , , and , with , we say that is congruent to modulo , or (mod ), if the difference is divisible 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.
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.