Can you do #3? **Module: Number Theory and Cryptography**---**Exercises on Modular Arithmetic and Number Theory**1. **Perform the following calculations:** - (a) Calculate \( 3^{340} \mod 341 \). - (b) Calculate \( 7^{89} \mod 100 \). - (c) Calculate \( 2^{10000} \mod 121 \).2. **Exponentiation Modulo m:** Implement the algorithm for exponentiation modulo \( m \) on a computer. Use this algorithm to check the result of Exercise 4.1.1.3. **Euler’s Theorem and the Chinese Remainder Theorem:** Use Euler's Theorem (and the Chinese Remainder Theorem) to show that \( n^{12} \equiv 1 \pmod{72} \) for all \( (n, 72) = 1 \).4. **Smallest Positive Integer λ:** What is the smallest positive integer \( \lambda \) such that \( n^{\lambda} \equiv 1 \pmod{100} \) for all \( (n, 100) = 1 \)?5. **Proving Modulo Congruence:** Prove that \( m^{\phi(n)} + n^{\phi(m)} \equiv 1 \pmod{mn} \) if \( (m, n) = 1 \).6. **Product of Distinct Primes:** Show that if \( n = pq \), where \( p \) and \( q \) are distinct primes, then \( a^{\phi(n) + 1} \equiv a \pmod{n} \) for all \( a \).7. **Composite Number Form:** Suppose that \( n = rs \) with \( r > 2 \), \( s > 2 \), and \( (r, s) = 1 \). Show that \( a^{\phi(n)/2} \equiv 1 \pmod{n} \), that is, \( a^{\phi(n)/2} \equiv 1 \pmod{n} \).8. **Product of Two Odd Primes:** Suppose \( n \) is the product of two odd primes, \( p \) and \( q \). Let

Name: Can you do #3? **Module: Number Theory and Cryptography**---**Exercise
Uploaded: 2024-03-20T07:06:41.000Z
Channel: Bartleby
Description: Can you do #3? **Module: Number Theory and Cryptography**---**Exercises on Modular Arithmetic and Number Theory**1. **Perform the following calculations:** - (a) Calculate \( 3^{340} \mod 341 \). - (b) Calculate \( 7^{89} \mod 100 \). - (c) Cal