Difference between revisions of "Modular arithmetic"
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== Introductory == | == Introductory == | ||
− | === | + | === Useful Facts === |
− | Consider four integers <math>{a},{b},{c},{d}</math> and a positive integer <math>{m}</math> such that <math>a\equiv b\pmod {m}</math> and <math>c\equiv d\pmod {m}</math>. In modular arithmetic, the following | + | Consider four integers <math>{a},{b},{c},{d}</math> and a positive integer <math>{m}</math> such that <math>a\equiv b\pmod {m}</math> and <math>c\equiv d\pmod {m}</math>. In modular arithmetic, the following [[identity | identities]] hold: |
* Addition: <math>a+c\equiv b+d\pmod {m}</math>. | * Addition: <math>a+c\equiv b+d\pmod {m}</math>. |
Revision as of 14:51, 20 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
[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.