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Fermat’s and Euler’s Theorem

Proof of Euler’s Theorem

Applications of Euler’s Theorem

Euler’s Theorem is generalization of fermat’s theorem. Here are the key differences between Fermat’s and Euler’s Theorem:

Feature Fermat’s Little Theorem Euler’s Theorem
Conditions Applicable when p is prime Applicable for any positive integer n and a relatively prime to n
Statement If p is prime and a is not divisible by p, then ap−1 ≡ 1 (mod p) If n is a positive integer and a is relatively prime to n, then aϕ(n) ≡ 1 (mod n)
Converse The converse is not always true. The converse is true: If aϕ(n) ≡ 1 (mod n), then a and n are coprime.
Requirement Requires the modulus p to be prime Does not require the modulus to be prime
Generalization Less general, restricted to prime moduli More general, applicable to any positive integer modulus

Euler’s Theorem

Euler’s Theorem states that for any integer a that is coprime with a positive integer m, the remainder of aϕ(m) divided by m is 1. We focus on proving Euler’s Theorem because Fermat’s Theorem is essentially a specific instance of it. This relationship arises because when p is a prime number, ϕ(p) equals p-1, thus making Fermat’s Theorem a subset of Euler’s Theorem under these conditions.

Euler’s theorem is a fundamental result in number theory, named after the Swiss mathematician Leonhard Euler. It states a relationship between the number theory functions and concepts of modular arithmetic. In this article, we will discuss Euler’s Theorem, including its statement and proof.

Table of Content

  • What is Euler’s Theorem?
  • Euler’s Theorem Formula
  • Historical Background of Euler’s Theorem
  • Proof of Euler’s Theorem
  • Applications of Euler’s Theorem
  • Euler’s Theorem Examples
  • Practice Questions on Euler’s Theorem

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What is Euler’s Theorem?

Euler’s Theorem is a fundamental concept in number theory. It states that if n and a are coprime positive integers, meaning that they have no mutual proper dividers other than m=1, then aϕ(n) and 1 are relative primes in modulo n....

Euler’s Theorem Formula

Statement of Euler’s Theorem can be used as formula for further calculations, i.e.,...

Historical Background of Euler’s Theorem

Euler’s theorem is named after the Swiss mathematician Leonhard Euler. Euler made numerous contributions to various branches of mathematics during the 18th century, and his work laid the groundwork for much of modern mathematics. Euler’s theorem specifically relates to modular arithmetic and the concept of totient function....

Proof of Euler’s Theorem

Let φ(n) = k, and let {a1, . . . , ak} be a reduced residue system mod n. For some ai in {a1, . . . , ak} Since (a, n) = 1, {aa1, . . . , aak} is another reduced residue system mod n. Since this is the same set of numbers mod n as the original system, the two systems must have the same product mod n: (aa1)· · ·(aak) = a1 · · · ak (mod n) ⇒ ak (a1 · · · ak) = a1 · · · ak (mod n) Now each ai is invertible mod n, so multiplying both sides by a1−1 · · · ak−1 , We get ak = 1 (mod n) or aφ(n) = 1 (mod n)....

Fermat’s and Euler’s Theorem

Euler’s Theorem is generalization of fermat’s theorem. Here are the key differences between Fermat’s and Euler’s Theorem:...

Applications of Euler’s Theorem

Euler’s Theorem has many applications in a wide range of areas, such as mathematics and even elsewhere. Here are some notable applications:...

Euler’s Theorem Examples

Example 1: Find the remainder when 3100 is divided by 7....

Practice Questions on Euler’s Theorem

Q1: Find the remainder when 250 is divided by 11....

FAQs on Euler’s Theorem

What is Euler’s Theorem?...

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Proof of Euler’s Theorem

Applications of Euler’s Theorem