Question 1: Prove that the sum of two even numbers is always even.

Answer: An even number can be written as 2k and 2m, so sum is 2k + 2m = 2(k + m), which is even.

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

Answer: An odd number is 2k + 1, even is 2m, product is (2k+1)*2m = 2m(2k+1) = 2( m(2k+1) ), which is even.

Question 3: Determine whether the number 29 is prime. Justify your answer.

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

Question 4: Construct a proof to show that any square of an odd number is odd.

Answer: Let an odd number be 2k+1. Its square is (2k+1)^2 = 4k^2 + 4k + 1 = 2(2k^2 + 2k) + 1, which is odd.

Question 5: If n is an odd prime greater than 2, prove that n is not divisible by 2.

Answer: Since n is odd, n = 2k + 1 for some integer k. If n were divisible by 2, n = 2m, which contradicts n = 2k + 1. Hence, n is not divisible by 2.

Question 6: A number is divisible by 3 if the sum of its digits is divisible by 3. Prove that if a number is divisible by 3, then its digit sum is divisible by 3.

Answer: Any number divisible by 3 can be expressed as 3k. The digit sum property follows from modular arithmetic, where the number mod 3 is equal to the sum of its digits mod 3, so if the number is divisible by 3, so is the digit sum.

Question 7: Show that the sum of an odd and an even number is odd.

Answer: Let odd = 2k+1, even = 2m. Sum = 2k+1 + 2m = 2(k + m) + 1, which is odd.

Question 8: Identify the common mistake in claiming that all prime numbers are odd, and correct it.

Answer: The mistake is ignoring 2, which is prime and even. Correct statement: 2 is prime and even; all other primes are odd.

Question 9: Challenge: Prove that the product of two odd numbers is odd.

Answer: Let odd numbers be 2k+1 and 2m+1. Their product is (2k+1)(2m+1) = 4km + 2k + 2m + 1 = 2(2km + k + m) + 1, which is odd.

Question 10: Explain why the statement ‘All prime numbers are odd’ is false, providing a counterexample.

Answer: The statement is false because 2 is a prime number that is even. Thus, not all primes are odd.