To find the remainder when \( 5^{40} \) is divided by 49, we can use properties from number theory, specifically Euler's theorem and modular arithmetic.

Euler's theorem states that for any two coprime integers \( a \) and \( n \), \( a^{\phi(n)} \equiv 1 \pmod{n} \), where \( \phi(n) \) is Euler's totient function.

1. **Calculate \( \phi(49) \)**:
   Since 49 is \( 7^2 \):
   \[
   \phi(49) = \phi(7^2) = 49 \left( 1 - \frac{1}{7} \right) = 49 \cdot \frac{6}{7} = 42
   \]
   So, \( \phi(49) = 42 \).

2. **Apply Euler's theorem**:
   Since \( 5 \) and \( 49 \) are coprime,
   \[
   5^{42} \equiv 1 \pmod{49}
   \]

3. **Relate \( 5^{40} \) to \( 5^{42} \)**:
   We know:
   \[
   5^{42} \equiv 1 \pmod{49}
   \]
   Thus, \( 5^{40} \cdot 5^2 \equiv 1 \pmod{49} \).

   Dividing both sides by \( 5^2 \) modulo 49,
   \[
   5^{40} \equiv (5^{-2} \pmod{49})
   \]

4. **Find the modular inverse \( 5^2 \) modulo 49**:
   \( 5^2 = 25 \). We need the modular inverse of 25 modulo 49.

   Using the Extended Euclidean Algorithm, we find that the modular inverse of 25 modulo 49 is 2:
   \[
   25 \cdot 2 \equiv 1 \pmod{49}
   \]

5. **Calculate \( 5^{40} \) modulo 49**:
   Thus,
   \[
   5^{40} \equiv 2 \pmod{49}
   \]

So, the remainder when \( 5^{40} \) is divided by 49 is \(\boxed{2}\).

Would you like more details or have any questions?

Here are 5 related questions that expand on this information:
1. How do you calculate the totient function \( \phi(n) \) for any given \( n \)?
2. What is the significance of Euler's theorem in number theory?
3. How can you find modular inverses using the Extended Euclidean Algorithm?
4. Can Euler's theorem be applied to non-coprime integers?
5. What are some practical applications of modular arithmetic in cryptography?

**Tip:** When working with large exponents in modular arithmetic, breaking the problem down using properties like Euler's theorem or the Chinese Remainder Theorem can simplify calculations significantly.

Learn how to find the remainder of 5^40 divided by 49 using number theory concepts like Euler's theorem, modular inverses, and the extended Euclidean algorithm. This problem covers advanced modular arithmetic suitable for university-level students.

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