Question 1: Prove that the sum of two even numbers is even. Use algebraic expressions for your proof.

Answer: Let an even number be 2a, another 2b. Sum: 2a + 2b = 2(a + b), which is divisible by 2, hence even.

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

Answer: Let odd = 2m + 1, even = 2n. Product: (2m + 1)(2n) = 4mn + 2n = 2(2mn + n), which is divisible by 2, so even.

Question 3: Identify whether 17 is prime and justify your answer.

Answer: 17 is prime because its only divisors are 1 and 17.

Question 4: Construct a proof that 2 is the only even prime number.

Answer: Any even number > 2 can be divided by 2, thus not prime. 2 is even and only divisible by 1 and itself, making it unique.

Question 1: Suppose you have an odd number and an even number. Prove that their sum is odd.

Answer: Let odd = 2p + 1, even = 2q. Sum: (2p + 1) + 2q = 2(p + q) + 1, which is odd because it is 2 times an integer plus 1.

Question 2: Prove that the sum of two odd numbers is even.

Answer: Let odd numbers be 2a + 1 and 2b + 1. Sum: 2a + 1 + 2b + 1 = 2(a + b + 1), which is divisible by 2, so even.

Question 3: Given a number n = 2k + 1, prove that n^2 is odd.

Answer: n^2 = (2k + 1)^2 = 4k^2 + 4k + 1 = 2(2k^2 + 2k) + 1, which is odd.

Question 1: A factory produces even-numbered batches of products. Prove that if the batch number is even, then the total number of products in that batch is divisible by 2.

Answer: Let batch number be 2m, total products per batch be 2m * p (p is number of items per batch). Since 2m * p = 2(m * p), the total is divisible by 2.

Question 1: Prove that if p is a prime greater than 2, then p is odd.

Answer: Assuming p is even and > 2, then p would be divisible by 2, contradicting primality. Therefore, p must be odd.

Question 2: Show that the product of two odd numbers is odd.

Answer: Odd numbers: 2a + 1, 2b + 1. Product: (2a + 1)(2b + 1) = 4ab + 2a + 2b + 1 = 2(2ab + a + b) + 1, which is odd.

Question 1: Determine whether 21 is prime or composite and justify.

Answer: 21 is composite because it has divisors 3 and 7 besides 1 and 21.

Question 2: Is 4 an odd or even number? Prove your answer.

Answer: 4 is even because it is divisible by 2, e.g., 4/2=2.

Question 3: Prove that any prime greater than 2 must be odd.

Answer: Assume prime p > 2 is even, then p divisible by 2, contradicting primality. So, p must be odd.

Question 1: A student claims: 'All odd numbers are prime.' Identify the mistake and correct the statement.

Answer: Mistake: Not all odd numbers are prime. For example, 9 is odd but not prime. Correct statement: 'Some odd numbers are prime, but not all.'

Question 2: A student states: 'The sum of two odd numbers is odd.' Find the error and explain.

Answer: Error: The sum of two odd numbers is even, not odd. For example, 3 + 5 = 8, which is even.