5. The CS 2050 office hours cubicle is moving! The new cubicle has a width of 3x and a length of y, where x, y ∈ Z+. Prove or disprove that the area of the cubicle is even whenever x is even. Make sure to include the introduction, body, and conclusion. Clearly state your reasoning for all statements and use a two-column proof for the body whenever possible.

6. Use a direct proof to show that if n + 9 is odd, then n2 −5n −14 is even. Make sure to include the introduction,body, and conclusion. Clearly state your reasoning for all statements and use a two-column proof for the body whenever possible.

7. Let n be an integer. Prove the statement “If 3n2 + 8 is even, then n is even.” Make sure to include the introduction,
body, and conclusion. Clearly state your reasoning for all statements and use a two-column proof for the body whenever
possible.

a) Prove the statement using a proof by contrapositive.
b) Prove the statement using a proof by contradiction.

8. Let p be the product of 5 distinct integers, where each of the 5 integers is between 1 and 63, inclusive. Prove or disprove
that if p is odd, then at least one of these 5 integers in its product must be odd. Make sure to include the introduction,
body, and conclusion. Clearly state your reasoning for all statements and use a two-column proof for the body whenever
possible.