301k - Homework 4
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 your scratch work.
1. Prove that if n is a perfect square, then n+2 is not a perfect
Use the following definition. A positive integer j is a perfect square if j=k^2 for some
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
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.