Can you do #5? ### Advanced Number Theory Problems and AlgorithmsThis section comprises a selection of problems and algorithms essential for those studying advanced number theory, specifically in algorithms related to modular exponentiation and properties of integer partitions. Each problem is designed to deepen your understanding and application of theoretical concepts.---**1. Evaluate Modular Exponentiation:**Compute the following using modular arithmetic: - (\(3^{340} \mod 341\)) - (\(7^{89} \mod 100\)) - (\(2^{1000} \mod 121\))**2. Implementing the Algorithm:**Implement the algorithm for exponentiation modulo \(m\) on a computer. Use this to verify the result of **Exercise 4.1.1.****3. Applications of Euler's Theorem:**Use Euler's Theorem (and the Chinese Remainder Theorem) to demonstrate that: - \(n^{12} \equiv 1 \mod 72\) for all \(n\) where \((n, 72) = 1\).**4. Finding the Smallest Positive Integer:**What is the smallest positive integer \(\lambda\) such that: - \(n^\lambda \equiv 1 \mod 100\) for all \((n, 100) = 1\)?**5. Proving Modular Exponential Relationships:**Prove that \(m^{\phi(n)} + \phi(m) \equiv 1 \mod mn\) if \((m, n) = 1\).**6. Showing Modular Exponential Properties:**Demonstrate that if \(n = pq\), a product of distinct primes, then: - \(a^{\phi(n) + 1} \equiv a \mod n\) for all \(a\).**7. Integer Partitions with Given Properties:**Suppose \(n = rs\) with \(r > 2\), \(s > 2\), and \((r, s) = 1\). Show that: - \(a^{\lambda(n)} \equiv 1 \mod n\) for all integers \(a\), satisfying \((a, n) = 1\).**8. Lambda Function for Two Odd Primes:**Suppose \(n\) is the product of two odd primes \(p\) and

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Description: Can you do #5? ### Advanced Number Theory Problems and AlgorithmsThis section comprises a selection of problems and algorithms essential for those studying advanced number theory, specifically in algorithms related to modular exponentiation and prope