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. | ||
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− | * | + | |
+ | == Intermediate == | ||
+ | === Topics === | ||
+ | * [[Fermat's Little Theorem]] | ||
+ | * [[Euler's Theorem]] | ||
+ | * [[Phi function]] | ||
+ | |||
=== See also === | === See also === |
Revision as of 19:12, 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.
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