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>. |
| + | |||
| + | == Introductory == | ||
=== Operations === | === Operations === | ||
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* Exponentiation: <math>a^e\equiv b^e\pmod {m}</math> where <math>e</math> is a positive integer. | * Exponentiation: <math>a^e\equiv b^e\pmod {m}</math> where <math>e</math> is a positive integer. | ||
| − | === | + | === Examples === |
* <math>{7}\equiv {1} \pmod {2}</math> | * <math>{7}\equiv {1} \pmod {2}</math> | ||
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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. | 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. | ||
| − | |||
| − | * | + | |
| + | == Intermediate == | ||
| + | === Topics === | ||
| + | * [[Fermat's Little Theorem]] | ||
| + | * [[Euler's Theorem]] | ||
| + | * [[Phi function]] | ||
| + | |||
=== See also === | === See also === | ||
Revision as of 18:12, 18 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.
Intermediate
Topics
