Math 511, Exam 1

Closed book. You are permitted a calculator and one 8½" ´ 11" sheet of handwritten notes (both sides). Be sure to show your work.

State the field laws.
Solve 27x + 11 = 43 (mod 322).
Solve x² + 6x + 3 = 0 (mod 29).
Given 361² = 2765² (mod 313121), factor 313121.
Compute 3²⁰⁵⁰ (mod 91).
Set up an RSA encryption system with primes p and q both over 50.
Which primes p (p £ 11) might divide Q(x) = x² - 4373?
Prove that if a ´ b = 0 with a ¹ 0, then b does not have a multiplicative inverse.
Short answer questions
- Is Z₂₅ a field? Explain why or why not.
- State two patterns you can find in the multiplication table for any field.
Explain Fermat's factoring method. Include both how and why the method works.