Number Theory - How to Prove n^7 is Congruent to n Mod 63

In summary, Euler's Theorem is a fundamental theorem in number theory that relates the greatest common divisor of two positive integers to their respective exponents in a power function. It is a generalization of Fermat's Little Theorem and uses Euler's totient function to determine whether a congruence relation holds. This theorem can be used to simplify modular arithmetic and has various real-world applications in fields such as cryptography and computer science.
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yeland404
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number theorm -- Euler theorem

Homework Statement



let be an integer that not divisible by 3. Prove that n^7[itex]\equiv[/itex]n mod 63

Homework Equations



none

The Attempt at a Solution


it is suffice to prove that n^7[itex]\equiv[/itex]n mod 7,n^7[itex]\equiv[/itex]n mod 9, i get
n^7[itex]\equiv[/itex]n mod 7 by Euler theorem , how to prove n^7[itex]\equiv[/itex]n mod 9
 
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Remember that Euler's totient function, [itex]\varphi (n)[/itex] is equal to the number of positive integers less than or equal to n that are coprime to n. What is [itex]\varphi (9)[/itex] and what does that imply by Euler's Theorem?
 

1. What is Euler's Theorem?

Euler's Theorem, also known as the Fermat-Euler Theorem, is a fundamental theorem in number theory that establishes a relationship between the greatest common divisor (GCD) of two positive integers and their respective exponents in a power function. It states that if a and n are relatively prime positive integers, then a^φ(n) ≡ 1 (mod n), where φ(n) is Euler's totient function which counts the number of positive integers less than n that are coprime to n.

2. How is Euler's Theorem related to Fermat's Little Theorem?

Euler's Theorem is a generalization of Fermat's Little Theorem, which states that if p is a prime number and a is any positive integer coprime to p, then a^(p-1) ≡ 1 (mod p). In other words, Euler's Theorem applies to any positive integer n, not just prime numbers like Fermat's Little Theorem.

3. What is the significance of Euler's Totient Function in Euler's Theorem?

Euler's Totient Function, denoted as φ(n), plays a crucial role in Euler's Theorem as it counts the number of positive integers less than n that are coprime to n. This function is essential in determining the value of a^φ(n) in the theorem, which is crucial in determining whether the congruence relation holds or not. It also has many other applications in number theory and cryptography.

4. Can Euler's Theorem be used to simplify modular arithmetic?

Yes, Euler's Theorem can be used to simplify modular arithmetic by reducing large exponents. For example, if we have the expression 23^101 ≡ x (mod 100), we can use Euler's Theorem to rewrite it as 23^φ(100) * 23 ≡ x (mod 100). Since φ(100) = 40, this becomes 23^40 * 23 ≡ x (mod 100). Using Euler's Theorem again, we know that 23^40 ≡ 1 (mod 100), so the expression simplifies to 1 * 23 ≡ x (mod 100), or simply 23 ≡ x (mod 100).

5. What are some real-world applications of Euler's Theorem?

Euler's Theorem has applications in many areas of mathematics, including number theory, cryptography, and coding theory. It is also used in the Euler's Phi function, which is used to generate keys in the RSA encryption algorithm. In addition, it has applications in fields such as computer science, engineering, and physics for solving problems involving modular arithmetic and number theory.

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