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>.
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'''Modular arithmetic''' is a special type of arithmetic that involves only [[integers]]. Since modular arithmetic is such a broadly useful tool in [[number theory]], we divide its explanations into several levels:
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* [[Modular arithmetic/Introduction|Introduction to modular arithmetic]]
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* [[Intermediate modular arithmetic]]
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* [[Olympiad modular arithmetic]]
  
=== Operations ===
 
  
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 operations are allowed:
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== Resources ==
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=== Introductory Resources ===
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==== Books ====
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* The AoPS [http://www.artofproblemsolving.com/Books/AoPS_B_Item.php?page_id=10 Introduction to Number Theory] by [[Mathew Crawford]].
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==== Classes ====
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* [http://www.artofproblemsolving.com/Classes/AoPS_C_ClassesS.php#begnum AoPS Introduction to Number Theory Course]
  
* Addition: <math>a+c\equiv b+d\pmod {m}</math>.
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=== Intermediate Resources ===
* Substraction: <math>a-c\equiv b-d\pmod {m}</math>.
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* [http://www.artofproblemsolving.com/Resources/Papers/SatoNT.pdf Number Theory Problems and Notes] by [[Naoki Sato]].
* Multiplication: <math>ac\equiv bd\pmod {m}</math>.
 
* Division: <math>\frac{a}{e}\equiv \frac{b}{e}\pmod {\frac{m}{\gcd(m,e)}}</math>, where <math>e</math> is a positive integer that divides <math>{a}</math> and <math>b</math>.
 
* Exponentiation: <math>a^e\equiv b^e\pmod {m}</math> where <math>e</math> is a positive integer.
 
  
=== Simple Examples ===
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=== Olympiad Resources ===
 
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* [http://www.artofproblemsolving.com/Resources/Papers/SatoNT.pdf Number Theory Problems and Notes] by [[Naoki Sato]].
* <math>{7}\equiv {1} \pmod {2}</math>
 
 
 
* <math>49^2\equiv 7^4\equiv (1)^4\equiv 1 \pmod {6}</math>
 
 
 
* <math>7a\equiv 14\pmod {49}\implies a\equiv 2\pmod {7}</math>
 
 
 
=== 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.
 
 
 
=== Examples ===
 
 
 
*
 
 
 
=== See also ===
 
 
 
* [[Number theory]]
 
* [[Quadratic residues]]
 

Latest revision as of 19:38, 19 July 2006

Modular arithmetic is a special type of arithmetic that involves only integers. Since modular arithmetic is such a broadly useful tool in number theory, we divide its explanations into several levels:


Resources

Introductory Resources

Books

Classes

Intermediate Resources

Olympiad Resources