189-235A: Algebra 1 Midterm Exam

Wednesday, October 25

Each of the four questions below is worth 25 points. No calculators or outside materials are allowed during the exam.

1. Let i be the complex number satisfying i^{2} = −1. Compute (1 + i)^{100}.

2. Let a and b be non-zero integers, and let L be the set of all strictly positive linear combinations of a and b:

L = {ra + sb, with r, s ∈ Z and ra + sb > 0}.

(a) Show that the smallest element of L divides a and b.

(b) Show that this smallest element is the gcd of a and b.

3. Compute the reduced residue modulo N (i.e., the unique integer 0 ≤ x ≤ N − 1 with x ≡ a (mod N)) of the integer

a = 7^{13198459348751983475867345892342398209234983465234531}

for the following values of N.

(a) N = 11;

(b) N = 5;

(c) N = 55.

4. Solve the following congruence equations (making sure you list all the distinct solutions in Z/NZ).

(a) 5x = 2 (mod 11).

(b) 10x = 4 (mod 22).

(c) 10x = 3 (mod 22).