Pries: 405 Number Theory
Sample Midterm: Spring 2012
1. Modular arithmetic:
A. Does x2 − 7y 2 = 3 have a solution with x, y ∈ Z? (Hint, pick a good modulus.)
B. Find the multiplicative inverse of 89 mod 101.
C. How many solutions x mod 7 · 11 · 5 are there to the equation x2 ≡ 4 mod 7 · 11 · 5?
(You do not need to find the solutions.)
2. Euler’s Theorem:
A. Find φ(200). What does this number represent?
B. Find a number x between 0 and 199 so that x ≡ 3243 mod 200.
3. Quadratic rings:
A. Factor 17 in Z[i].
√
B. Write down an example of the failure of unique factorization in Z[ −13].
4. Use quadratic reciprocity rules to determine whether each congruence has a solution.
A. x2 ≡ −1 mod 103.
B. x2 ≡ 2 mod 103.
C. x2 ≡ 35 mod 103.
5. Primitive roots:
A. Find a primitive root modulo 13. Use it to find all solutions to x3 ≡ 1 mod 13.
B. Suppose p ≡ 1 mod 3. Prove there are 3 solutions to x3 ≡ 1 mod p.
C. Let q be a prime number such that p = 2q + 1 is also a prime number. Suppose
q ≡ 1 mod 4. Show that 2 is a primitive root modulo p.
6. Proofs:
A. Write down the main steps to prove Fermat’s Little Theorem and Euler’s criterion.
B. Write a detailed proof (Prime Claim): if p is a prime and p | xy then p | x or p | y.
C. Write a detailed proof: if x2 ≡ 1 mod p then x ≡ ±1 mod p.
D. If p ≡ 1 mod 4, prove that p is not prime in Z[i]. Hint: use −1 ≡ x2 mod p.