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The Problem Solver's Resource

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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 $a|c$ .

Diverging-Converging Theorem

A series $\sum_{i=0}^{\infty}S_i$ converges iff $\lim S_i=0$ .

Errata

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

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