Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

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

(topic)
(this belongs in limits...)
Line 34: Line 34:
 
If <math>a|bc</math> and <math>(a,b) = 1</math>, then <math>a|c</math>.
 
If <math>a|bc</math> and <math>(a,b) = 1</math>, then <math>a|c</math>.
  
−
===Diverging-Converging Theorem===
 
−
A series <math>\sum_{i=0}^{\infty}S_i</math> converges iff <math>\lim S_i=0</math>.
 
  
 
===Errata===
 
===Errata===

Revision as of 12:12, 23 November 2007



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

Number Theory

This section covers number theory, especially modulos (moduli?).

Definitions

  • $n\equiv a\pmod{b}$ if $n$ is the remainder when $a$ is divided by $b$ to give an integral amount.
  • $a|b$ (or $a$ divides $b$) if $b=ka$ for some integer $k$.

Special Notation

Occasionally, if two equivalent expressions are both modulated by the same number, the entire equation will be followed by the modulo.

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

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

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 $p$, $(p-1)! \equiv -1 \pmod p$.

Fermat-Euler Identitity

If $gcd(a,m)=1$, then $a^{\phi{m}}\equiv1\pmod{m}$, where $\phi{m}$ is the number of relatively prime numbers lower than $m$.

Gauss's Theorem

If $a|bc$ and $(a,b) = 1$, then $a|c$.


Errata

All quadratic residues are $0$ or $1\pmod{4}$and $0$, $1$, or $4$ $\pmod{8}$.

Back to page 5 | Continue to page 7



Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=User:Temperal/The_Problem_Solver%27s_Resource6&oldid=19875"