# Difference between revisions of "Euler's Totient Theorem"

m (Euler's totient theorem moved to Euler's Totient Theorem: Capitalization policy is currently to capitalize names of theorems, I believe (see [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=97741 here]).) |
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− | + | '''Euler's Totient Theorem''' is a theorem closely related to his [[totient function]]. | |

− | Let <math>\phi(n)</math> be [[Euler's totient function]]. If <math>{a}</math> is an integer and <math>m</math> is a positive integer [[relatively prime]] to <math>a</math>, | + | == Theorem == |

+ | Let <math>\phi(n)</math> be [[Euler's totient function]]. If <math>n</math> is a positive integer, <math>\phi{(n)}</math> is the number of integers in the range <math>\{1,2,3\cdots{,n}\}</math> which are relatively prime to <math>n</math>. If <math>{a}</math> is an integer and <math>m</math> is a positive integer [[relatively prime]] to <math>a</math>,Then <math>{a}^{\phi (m)}\equiv 1 \pmod {m}</math>. | ||

− | + | == Credit == | |

− | This theorem is credited to [[Leonhard Euler]]. It is a generalization of [[Fermat's Little Theorem]], which specifies | + | This theorem is credited to [[Leonhard Euler]]. It is a generalization of [[Fermat's Little Theorem]], which specifies it when <math>{m}</math> is prime. For this reason it is also known as Euler's generalization or the Fermat-Euler theorem. |

− | === See also | + | ==Proof== |

+ | |||

+ | Consider the set of numbers <math>A = \{ n_1, n_2, ... n_{\phi(m)} \} \pmod{m}</math> such that the elements of the [[set]] are the numbers relatively [[prime]] to <math>m</math>. | ||

+ | It will now be proved that this set is the same as the set <math>B = \{ an_1, an_2, ... an_{\phi(m)} \} \pmod{m}</math> where <math> (a, m) = 1</math>. All elements of <math>B</math> are relatively prime to <math>m</math> so if all elements of <math>B</math> are distinct, then <math>B</math> has the same elements as <math>A.</math> In other words, each element of <math>B</math> is congruent to one of <math>A</math>.This means that <math> n_1 n_2 ... n_{\phi(m)} \equiv an_1 \cdot an_2 ... an_{\phi(m)}</math><math>\pmod{m}</math> → <math>a^{\phi (m)} \cdot (n_1 n_2 ... n_{\phi(m)}) \equiv n_1 n_2 ... n_{\phi(m)}</math><math>\pmod{m}</math> → <math>a^{\phi (m)} \equiv 1</math><math>\pmod{m}</math> as desired. Note that dividing by <math> n_1 n_2 ... n_{\phi(m)}</math> is allowed since it is relatively prime to <math>m</math>. | ||

+ | |||

+ | ==Problems== | ||

+ | |||

+ | ===Introductory=== | ||

+ | *(BorealBear) Find the last two digits of <math> 7^{81}-3^{81} </math>. ([[Euler's Totient Theorem Problem 1 Solution|Solution]]) | ||

+ | *(BorealBear) Find the last two digits of <math> 3^{3^{3^{3}}} </math>. ([[Euler's Totient Theorem Problem 2 Solution|Solution]]) | ||

+ | *([[2018 AMC 10B Problems/Problem 16|2018 AMC 10B]]) Let <math>a_1,a_2,\dots,a_{2018}</math> be a strictly increasing sequence of positive integers such that <cmath>a_1+a_2+\cdots+a_{2018}=2018^{2018}.</cmath> | ||

+ | What is the remainder when <math>a_1^3+a_2^3+\cdots+a_{2018}^3</math> is divided by <math>6</math>? | ||

+ | |||

+ | <cmath>\textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 4</cmath> | ||

+ | *([[1983 AIME Problems/Problem 6|1983 AIME]]) Let <math>a_n=6^{n}+8^{n}</math>. Determine the remainder upon dividing <math>a_ {83}</math> by <math>49</math>. | ||

+ | |||

+ | == See also == | ||

* [[Number theory]] | * [[Number theory]] | ||

* [[Modular arithmetic]] | * [[Modular arithmetic]] | ||

* [[Euler's totient function]] | * [[Euler's totient function]] | ||

+ | * [[Carmichael function]] | ||

+ | |||

+ | [[Category:Number theory]] | ||

+ | |||

+ | [[Category:Theorems]] |

## Latest revision as of 14:12, 23 April 2021

**Euler's Totient Theorem** is a theorem closely related to his totient function.

## Theorem

Let be Euler's totient function. If is a positive integer, is the number of integers in the range which are relatively prime to . If is an integer and is a positive integer relatively prime to ,Then .

## Credit

This theorem is credited to Leonhard Euler. It is a generalization of Fermat's Little Theorem, which specifies it when is prime. For this reason it is also known as Euler's generalization or the Fermat-Euler theorem.

## Proof

Consider the set of numbers such that the elements of the set are the numbers relatively prime to . It will now be proved that this set is the same as the set where . All elements of are relatively prime to so if all elements of are distinct, then has the same elements as In other words, each element of is congruent to one of .This means that → → as desired. Note that dividing by is allowed since it is relatively prime to .

## Problems

### Introductory

- (BorealBear) Find the last two digits of . (Solution)
- (BorealBear) Find the last two digits of . (Solution)
- (2018 AMC 10B) Let be a strictly increasing sequence of positive integers such that

What is the remainder when is divided by ?

- (1983 AIME) Let . Determine the remainder upon dividing by .