CS
301k - Homework 5

Answer all questions clearly. Your solution set must be stapled together if you use more than one piece of paper. 100 points are possible. Please put your name and EID in the top left corner of the first page of your solution set.

For questions 1, 2, 5 and 6, write the claim in logical notation before you do the proofs, , show how you are applying universal generalization, write down your givens and goal, and show your scratch work (and your final proof, of course). For the other questions, you can give the final proof only.

1. Prove that if n is a perfect square, then n+2 is not a perfect square.

Use the following definition. A positive integer j is a perfect square if j=k^2 for some integer k.

2. Prove that if n is an integer and 3n+2 is even, then n is even.

a) Do a proof by contraposition

b) Do a proof by contradiction

3. Prove there is a positive integer that equals the sum of the positive integers that are not greater than it.

4. Suppose that you have a wheel of fortune that has the integers from 1 to 36 painted on it in a random way. Show that regardless of the ordering of the numbers, there are three consecutive (on the wheel) numbers with a sum of 55 or greater. Hint: Think of two different ways to compute the sum of the numbers on the wheel, and then compare the two quantities you get.

5. Use a direct proof. Prove: The sum of three odd integers is odd.

6. Prove: The product of any two even integers is divisible by 4. (Do a direct proof).

7. Prove: The sum of any three consecutive integers is divisible by 3. Note: Two integers are consecutive if one is one more than the other.