MIDTERM EXAM MATH 115: NUMBER THEORY

Answer each question completely, and give sufficient justification and proof. Write neatly and in complete sentences!

Name Student ID

Problem 1  /10
Problem 2  /10
Problem 3  /15
Problem 4  /10
Problem 5 (Bonus)  /5
Total Score  /45

Midterm Grade Problem 1.

Date: July 15, 2004.

(a) Compute g = gcd(2004, 99).

(b) For g above, find integers x, y ∈ ℤ such that 2004x + 99y = g.

Problem 2. Let a, b ∈ ℤ_{>1} satisfy a^{3} = b^{2}. Show that there exists a d ∈ ℤ such that a = d^{2} and b = d^{3}.

4 115 MIDTERM

Problem 3.

(a) Find a solution x ∈ ℤ/27ℤ to the congruence x^{2} − 7x ≡ 6 (mod 27).

(b) How many distinct solutions x ∈ ℤ/243ℤ are there to the con-

gruence

x^{2} − 7x ≡ 6 (mod 243)?

115 MIDTERM 5

Problem 4. What is the smallest prime divisor of n = 365^{2004} + 94?

6 115 MIDTERM

Problem 5 (Bonus). Let α = arctan(7/2). Show that sin(α) is irrational.