Difference between revisions of "User:Temperal/The Problem Solver's Resource6"

(Properties)
(Useful Theorems)
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==Useful Theorems==
 
==Useful Theorems==
 
Fermat's Little Theorem:For a prime <math>p</math> and a number <math>a</math> such that <math>a\ne{p}</math>, <math>a^{p-1}\equiv 1 (\pmod{p}</math>.
 
Fermat's Little Theorem:For a prime <math>p</math> and a number <math>a</math> such that <math>a\ne{p}</math>, <math>a^{p-1}\equiv 1 (\pmod{p}</math>.
 +
 
Wilson's Theorem: For a prime <math>p</math>, <math> (p-1)! \equiv -1 (mod p)</math>.
 
Wilson's Theorem: For a prime <math>p</math>, <math> (p-1)! \equiv -1 (mod p)</math>.

Revision as of 21:09, 5 October 2007



The Problem Solver's Resource
Introduction | Other Tips and Tricks | Methods of Proof | You are currently viewing page 6.

Modulos

Definition

  • $n\equiv a\pmod{b}$ if $n$ is the remainder when $a$ is divided by $b$ to give an integral amount.

Special Notation

Properties

For any number there will be only one congruent number modulo $m$ between $0$ and $m-1$.

If $a\equiv b \pmod{m}$ and $c \equiv d \pmod{m}$, then $(a+c) \equiv (b+d) \pmod {m}$.


$a \pmod{m} + b \pmod{m} \equiv (a + b) \pmod{m}$

$a \pmod{m} - b \pmod{m} \equiv (a - b) \pmod{m}$

$a \pmod{m} \cdot b \pmod{m} \equiv (a \cdot b) \pmod{m}$

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Useful Theorems

Fermat's Little Theorem:For a prime $p$ and a number $a$ such that $a\ne{p}$, $a^{p-1}\equiv 1 (\pmod{p}$.

Wilson's Theorem: For a prime $p$, $(p-1)! \equiv -1 (mod p)$.