You may use one sheet of handwritten notes, but no other sources. The exam consists of five questions, and lasts fifty minutes. Please work all five problems. Please show all work clearly and in reasonable detail. Answers without appropriate supporting work or requested explanations may not receive full credit. No calculators are allowed. Please ring your section below:

1. (20 points) A certain island is populated by only two types of people: knights, who always tell the truth, and knaves, who always lie. Two of the inhabitants of the island tell you the following:

   A says: Both of us are knaves.
   B says nothing.

   What are A and B? (Note: make sure that you show that your answer is the only consistent solution.)

2. (20 points) Solve the congruence 4x ≡ 5 (mod 9).
3. 1. (10 points) Convert (100110110011)₂ into its hexadecimal expansion.
   2. (10 points) How many zeroes are there at the end of the number 35⁶ × 44²?
4. (20 points) Recall that A×B denotes the Cartesian product of the two sets A and B. Let A, B, and C be three sets. Prove that

5. (20 points) Prove that the square of any positive odd integer has the form 8m + 1 for some integer m.