Question 1: Prove that the sum of two even numbers is always even. Write a general proof.

Answer: Let the two even numbers be 2a and 2b, where a and b are integers. Their sum is 2a + 2b = 2(a + b), which is divisible by 2, so the sum is even.

Question 2: Identify and correct the mistake in the following proof: 'All odd numbers can be written as 2k + 1. The sum of two odd numbers is (2k + 1) + (2m + 1) = 2(k + m + 1), which is even. Therefore, the sum of two odd numbers is even.'

Answer: The mistake is in the calculation: (2k + 1) + (2m + 1) = 2k + 2m + 2 = 2(k + m + 1). Actually, this is correct, but the statement about the sum being even is true. The error lies in the misconception that sum of two odd numbers is always even, which is actually correct, so students must identify that the proof is correct or that the initial claim is false if they spot other errors.

Question 3: Show whether 17 is prime or not, and explain your reasoning.

Answer: 17 is prime because it has no divisors other than 1 and itself.

Question 4: Construct a proof to show that the product of two odd numbers is odd.

Answer: Let the two odd numbers be 2a + 1 and 2b + 1, where a and b are integers. Their product is (2a + 1)(2b + 1) = 4ab + 2a + 2b + 1 = 2(2ab + a + b) + 1, which is of the form 2k + 1, so the product is odd.

Question 5: Is the statement 'All prime numbers are odd' true? Justify your answer with proof or counterexample.

Answer: No, the statement is false because 2 is a prime number and it is even. All other primes are odd, but 2 is an exception.

Question 6: Identify the common misconception in the following statement: 'Since 3 is prime, all numbers divisible by 3 are also prime.'

Answer: The misconception is that divisibility by 3 does not imply the number is prime; for example, 9 is divisible by 3 but not prime.

Question 7: Use a proof by contradiction to show that an odd number squared is odd.

Answer: Assume the contrary: that an odd number n squared is even. Then, n = 2k + 1 for some integer k. n² = (2k + 1)² = 4k² + 4k + 1 = 2(2k² + 2k) + 1, which is odd. This contradicts the assumption that n² is even, so n² must be odd.

Question 8: Determine whether 29 is prime or composite and justify your answer.

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

Question 9: Prove that the product of an even and an odd number is even.

Answer: Let the even number be 2a and the odd number be 2b + 1. Their product is 2a(2b + 1) = 4ab + 2a = 2(2ab + a), which is divisible by 2, so the product is even.

Question 10: Identify and explain the mistake in the following reasoning: 'Since 2 is prime, all multiples of 2 are prime.'

Answer: The mistake is that multiples of 2 greater than 2 are not prime; for example, 4, 6, 8 are multiples of 2 but not prime.