Euler's Theorem is a fundamental result in number theory that states:

If $a$ and $n$ are two positive integers such that $\gcd(a, n) = 1$ (i.e., $a$ and $n$ are coprime), then:

$a^{\phi(n)} \equiv 1 \pmod{n}$

where $\phi(n)$ is Euler's totient function, which counts the number of positive integers less than $n$ that are coprime with $n$.

Key Points:

• Coprime Condition: The condition $\gcd(a, n) = 1$ is essential for the theorem to hold.
• Totient Function: The function $\phi(n)$ plays a crucial role in the theorem. It is defined as the number of integers from 1 to $n$ that are relatively prime to $n$.

Examples:

1. If $n = 7$:

   • $\phi(7) = 6$ (since 7 is prime, $\phi(7) = 7 - 1$).
   • For $a = 3$, which is coprime to 7: $3^6 \equiv 1 \pmod{7}.$

2. If $n = 10$:

   • $\phi(10) = 4$ (since $\phi(10) = \phi(2) \times \phi(5) = 1 \times 4$).
   • For $a = 3$, which is coprime to 10: $3^4 \equiv 1 \pmod{10}.$

Importance:

Euler's Theorem is a generalization of Fermat's Little Theorem. It is particularly useful in the fields of cryptography, especially in the RSA encryption algorithm.

Would you like to know more about Euler's theorem or its applications? Here are some related questions:

1. How is Euler's Theorem used in the RSA encryption algorithm?
2. What is Euler's totient function, and how is it computed for any integer $n$?
3. How does Euler's Theorem generalize Fermat's Little Theorem?
4. Can Euler's Theorem be extended to non-coprime $a$ and $n$?
5. What are the proofs or derivations of Euler's Theorem?

Tip: When working with Euler's Theorem, always check that the integers are coprime before applying the theorem.