Question 1: Prove that the sum of two even numbers is even. Use algebraic expressions to justify your answer.

Answer: Let even numbers be 2a and 2b; sum = 2a + 2b = 2(a + b), which is divisible by 2, so sum is even.

Question 2: Show that the product of an odd number and an even number is even using algebra.

Answer: Let odd = 2a + 1, even = 2b; product = (2a + 1)×2b = 2b(2a + 1) = 2(2ab + b), which is divisible by 2, so product is even.

Question 3: Determine whether the number 29 is prime. State your reason.

Answer: 29 is prime because it has no divisors other than 1 and 29.

Question 4: Construct an algebraic proof to show that the square of an odd number is odd.

Answer: Let odd = 2a + 1; then (2a + 1)^2 = 4a^2 + 4a + 1 = 2(2a^2 + 2a) + 1, which is odd because it is 2k + 1.

Question 5: If p is a prime greater than 2, prove that p is odd.

Answer: Suppose p is even and prime; then p = 2k for some integer k. Since p > 2, k ≥ 2, so p is divisible by 2 and at least 4, contradicting primality. Therefore, p must be odd.

Question 6: Prove that the product of two odd numbers is odd.

Answer: Let odd = 2a + 1 and 2b + 1; then product = (2a + 1)(2b + 1) = 4ab + 2a + 2b + 1 = 2(2ab + a + b) + 1, which is odd.

Question 7: Determine if the number 51 is prime. Justify your answer.

Answer: 51 is not prime because it is divisible by 3 (since 5 + 1 = 6, divisible by 3).

Question 8: Show algebraically that the sum of an odd number and an even number is odd.

Answer: Let odd = 2a + 1, even = 2b; sum = (2a + 1) + 2b = 2a + 2b + 1 = 2(a + b) + 1, which is odd.

Question 9: Identify the mistake in this reasoning: "Since 2 is prime, all even numbers are prime."

Answer: The mistake is that not all even numbers are prime; only 2 is prime, while other even numbers (like 4, 6) are composite.

Question 10: Prove that for any prime number p > 2, p^2 - 1 is divisible by 6.

Answer: Since p > 2 and prime, p is odd, so p = 2k + 1. Then p^2 - 1 = (2k + 1)^2 - 1 = 4k^2 + 4k = 4k(k + 1). Since k and k + 1 are consecutive integers, one is even, so product is divisible by 2. Also, p is odd, so p^2 ≡ 1 mod 3, so p^2 - 1 ≡ 0 mod 3. Combining, p^2 - 1 is divisible by 2 and 3, hence divisible by 6.