# Expert Answers

1. Use a direct proof to show that the sum of two even integers is even.

3. Show that the additive inverse, or negative, of an even number is an even number using a direct proof.

4. Use a direct proof to show that the product of two odd numbers is odd.

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

6. Use a direct proof to show that the product of two rational numbers is rational.

7. Prove that if m and n are integers and mn is even, then m is even or n is even.

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

a) a proof by contraposition

b) a proof by contradiction

9. Prove the proposition P(1), where P(n) is the proposition “If n is a positive integer, then n2 ≥ n”. What kind of proof did you use?

11. Show that if you pick three socks from a drawer containing just blue socks and black socks, you must get either a pair of blue socks or a pair of black socks.

12. Prove that if n is a positive integer, then n is even if and only if 7n + 4 is even.

13. Find a counterexample to the statement that every positive integer can be written as the sum of the squares of three integers.

3. Show that the additive inverse, or negative, of an even number is an even number using a direct proof.

4. Use a direct proof to show that the product of two odd numbers is odd.

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

6. Use a direct proof to show that the product of two rational numbers is rational.

7. Prove that if m and n are integers and mn is even, then m is even or n is even.

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

a) a proof by contraposition

b) a proof by contradiction

9. Prove the proposition P(1), where P(n) is the proposition “If n is a positive integer, then n2 ≥ n”. What kind of proof did you use?

11. Show that if you pick three socks from a drawer containing just blue socks and black socks, you must get either a pair of blue socks or a pair of black socks.

12. Prove that if n is a positive integer, then n is even if and only if 7n + 4 is even.

13. Find a counterexample to the statement that every positive integer can be written as the sum of the squares of three integers.

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