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

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After multiplying through by <math>23!</math>, we know that every term on the left-hand-side will be divisible by 13 except for <math>\frac{23!}{13}</math>. We wish to find the remainder when <math>23\cdot22\cdot21...\cdot1</math> is divided by 13. From Wilson's Theorem, we know that <math>12!\equiv-1\pmod{13}</math> so we consider (mod 13). Thus, the remainder is <math>10\cdot9\cdot8...\cdot1\cdot12!\equiv10!\cdot{-1}\pmod{13}</math> which comes out to be 7. Thus, our answer is 7.
 
After multiplying through by <math>23!</math>, we know that every term on the left-hand-side will be divisible by 13 except for <math>\frac{23!}{13}</math>. We wish to find the remainder when <math>23\cdot22\cdot21...\cdot1</math> is divided by 13. From Wilson's Theorem, we know that <math>12!\equiv-1\pmod{13}</math> so we consider (mod 13). Thus, the remainder is <math>10\cdot9\cdot8...\cdot1\cdot12!\equiv10!\cdot{-1}\pmod{13}</math> which comes out to be 7. Thus, our answer is 7.
  
−
===Fermat-Euler Identitity===
+
===Euler's Phi Theorem===
 
If <math>gcd(a,m)=1</math>, then <math>a^{\phi{m}}\equiv1\pmod{m}</math>, where <math>\phi{m}</math> is the number of relatively prime  numbers lower than <math>m</math>.
 
If <math>gcd(a,m)=1</math>, then <math>a^{\phi{m}}\equiv1\pmod{m}</math>, where <math>\phi{m}</math> is the number of relatively prime  numbers lower than <math>m</math>.
−
 
  
 
===Errata===
 
===Errata===

Revision as of 21:38, 10 January 2009

Introduction | Other Tips and Tricks | Methods of Proof | You are currently viewing page 6.

Number Theory

This section covers number theory, specifically Fermat's Little Theorem, Wilson's Theorem, and Euler's Totient Theorem.

Definitions

  • $n\equiv a\pmod{b}$ if $n$ is the remainder when $a$ is divided by $b$ to give an integral amount. Also, this means b divides (n-a).
  • $a|b$ (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.

$(a_1, a_2,...a_n)$ refers to the greatest common factor of $a_1, a_2, ...a_n$ and $[a_1, a_2, ...a_n]$ refers to the lowest common multiple of $a_1, a_2,...a_n$.

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 $(a,b)=1$, $a^{p-1}\equiv 1 \pmod{p}$. A frequently used result of this is $a^p\equiv a\pmod{p}$.

Example Problem 1

Find all primes p such that $p|2^p+1$.

Solution

Firstly, p=2 clearly does not work. Now, as all other primes are odd, $(2, p)=1$ and hence $2^p\equiv2\pmod{p}$. After adding one, we have $3\equiv0\pmod{p}$ since p divides $2^p+1$. However, that means p must divide 3, so the only prime possible is 3. Indeed, $2^3+1=9$ is a multiple of 3.

Wilson's Theorem

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

Example Problem 2=

Let $a$ be an integer such that $\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{23}=\frac{a}{23!}$. Find the remainder when $a$ is divided by $13$.

Solution

After multiplying through by $23!$, we know that every term on the left-hand-side will be divisible by 13 except for $\frac{23!}{13}$. We wish to find the remainder when $23\cdot22\cdot21...\cdot1$ is divided by 13. From Wilson's Theorem, we know that $12!\equiv-1\pmod{13}$ so we consider (mod 13). Thus, the remainder is $10\cdot9\cdot8...\cdot1\cdot12!\equiv10!\cdot{-1}\pmod{13}$ which comes out to be 7. Thus, our answer is 7.

Euler's Phi Theorem

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

Errata

An integer n is a quadratic residue (mod m) if and only if there exists an integer p such that $p^2\equiv n\pmod{m}$. All quadratic residues are $0$ or $1\pmod{4}$and $0$, $1$, or $4$ $\pmod{8}$. All cubic residues (mod 9) are 0, 1, or -1.

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