Euler’s Theorem Formula
Statement of Euler’s Theorem can be used as formula for further calculations, i.e.,
aϕ(n) ≡ 1 (mod n)
Where,
- a is any integer coprime to n,
- n is a positive integer,
- ϕ(n) is Euler’s totient function, and
- ≡ denotes equivalence,
- mod n represents congruence modulo n.
Example Showing Euler’s Theorem Formula
Problem: Verify Euler’s Theorem for a = 3 and n = 8.
Solution:
First, we calculate ϕ(8). The numbers less than 8 that are coprime to 8 are 1, 3, 5, and 7. Thus, ϕ(8)=4.
Next, calculate 34 and find its remainder when divided by 8
34 = 81
Now, find 81 mod 8
81 mod 8 ≡ 1
Thus, 34 ≡ 1 (mod 8), which verifies Euler’s Theorem.
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