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Pries: 405 Number Theory, Spring 2012. Homework 3. Due: Friday 2/10. Modular arithmetic and Euler’s Theorem Read: Stein sections 2.1, 2.2 and 2.3. Do 8 of the following problems. 1. Stein 2.1, 2.5, 2.11, 2.13, 2.18, 2.23, 2.30, 2.33. 2. An element a ∈ Z/m is a zero-divisor if a 6≡ 0 mod m and ax ≡ 0 mod m for some x 6≡ 0 mod m. Find the zero-divisors in Z/20. Prove that a ∈ Z/m is a zero-divisor if and only if gcd(a, m) 6= 1. 3. In class, we proved that there is a solution to ax ≡ 1 mod m if and only if gcd(a, m) = 1. Prove that there is a solution to ax ≡ c mod b if and only gcd(a, m) divides c. 4. For which primes p, is there a solution to x2 ≡ 2 mod p? Generate some data and make a conjecture. 5. What is φ(2012)? John Louis von Neumann If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is. 1