Question 1: Prove that the sum of two even numbers is always even. Use algebra to support your answer.

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

Question 2: Show that the difference between two odd numbers is always even using algebraic proof.

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

Question 3: Using algebra, prove that an odd number multiplied by an even number results in an even number.

Answer: Let odd = 2a + 1, even = 2b. Product = (2a + 1) * 2b = 4ab + 2b = 2(2ab + b), which is divisible by 2, hence even.

Question 4: Determine whether the number 17 is prime, and justify your answer using divisibility rules.

Answer: 17 has no divisors other than 1 and itself, so 17 is prime.

Question 5: Prove algebraically that the sum of two odd primes cannot be prime.

Answer: Let odd primes be 2a + 1 and 2b + 1. Sum = 2a + 2b + 2 = 2(a + b + 1), which is even and > 2, so not prime.

Question 6: A number is considered 'special' if it is prime and odd. Given that 29 is a prime number, explain if 29 fits this criteria and why.

Answer: Yes, 29 is prime and odd, so it fits the criteria of a 'special' number.

Question 7: Construct a proof to show that the product of an even number and an odd number is always even.

Answer: Let even = 2a, odd = 2b + 1. Product = 2a*(2b + 1) = 4ab + 2a = 2(2ab + a), which is divisible by 2, hence even.

Question 8: Explain why the number 15 is not prime using algebraic factors.

Answer: 15 = 3 * 5, both greater than 1, so 15 is composite, not prime.

Question 9: Identify any common mistakes students might make when proving that the sum of two even numbers is even, and explain how to avoid them.

Answer: A common mistake is to forget that the sum of two even numbers is always divisible by 2, or to use specific numbers without general proof. To avoid this, students should use algebraic expressions to show the property holds for all even numbers.

Question 10: Challenge: Prove that if a number is divisible by 4, it is also divisible by 2.

Answer: Let n be divisible by 4, so n = 4k for some integer k. Since 4k = 2*(2k), n is divisible by 2.