Can you do #10? ### Number Theory Problem Set1. **Question**: What is the smallest positive integer \( n \) such that \( \phi(n) = 12 \) and \( (n, 100) = 1 \)?2. **Question**: Prove that \( m^{\phi(n)} + n^{\phi(m)} \equiv 1 \pmod{\operatorname{mn}} \) if \( (m,n) = 1 \).3. **Question**: Show that if \( n = pq \), a product of distinct primes, then \( a^{\phi(n)+1} \equiv a \pmod{n} \) for all \( a \).4. **Question**: Suppose that \( n = rs \) with \( r > 2 \), \( s > 2 \), and \( (r,s) = 1 \). Show that \( a^{\frac{\phi(n) + \phi(a)}{2}} \equiv 1 \pmod{n} \), that is, \( a^{\frac{\phi(n)}{2} + 1} \equiv 1 \pmod{n} \).5. **Question**: Suppose \( n \) is the product of two odd primes, \( p \) and \( q \). Let \[ \lambda(n) = \frac{(p-1)(q-1)}{(p-1, q-1)}. \] Show that \( a^{\lambda(n)} \equiv 1 \pmod{n} \) for all integers \( a \), satisfying \( (a, n) = 1 \).6. **Question**: Use a computer to find the composite numbers \( n \leq 2000 \) such that \( 2^n \equiv 2 \pmod{n} \). Repeat the exercise to find composite \( n \) such that \( 3^n \equiv 3 \pmod{n} \).7. **Question**: Prove that \( a^{560} \equiv 1 \pmod{561} \) for all \( (a, 561) = 1 \).8. **Question**: Suppose \( a^x \equiv 1 \pmod{m} \) and \( a^

Name: Can you do #10? ### Number Theory Problem Set1. **Question**: What is
Uploaded: 2024-03-20T07:06:41.000Z
Channel: Bartleby
Description: Can you do #10? ### Number Theory Problem Set1. **Question**: What is the smallest positive integer \( n \) such that \( \phi(n) = 12 \) and \( (n, 100) = 1 \)?2. **Question**: Prove that \( m^{\phi(n)} + n^{\phi(m)} \equiv 1 \pmod{\operatorname{mn}}