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

Answer: Because even numbers can be written as 2k and 2m, their sum is 2k + 2m = 2(k + m), which is divisible by 2, hence even.

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

Answer: Let odd = 2n + 1 and even = 2m, then their product is (2n + 1)*2m = 4nm + 2m = 2(2nm + m), which is divisible by 2, hence even.

Question 3: Determine whether the number 49 is prime or composite. Show your reasoning.

Answer: 49 = 7 × 7, so it is composite.

Question 4: Construct a number on the grid that is both odd and prime.

Answer: Number 3 or 5 or 7 (any of these are odd primes).

Question 1: Prove that the sum of an odd and an even number is always odd.

Answer: Let odd = 2n + 1 and even = 2m, then sum = (2n + 1) + 2m = 2(n + m) + 1, which is odd.

Question 2: Show that the square of an odd number is odd.

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

Question 3: Identify if the number 51 is prime. If not, give its prime factors.

Answer: 51 = 3 × 17, so not prime.

Question 4: Construct a number on the grid that is both even and composite.

Answer: Any even number greater than 2, e.g., 4, 6, 8, etc.

Question 1: A school is arranging students in pairs for a sports event. Prove that if the total number of students is odd, then at least one student will be left without a partner.

Answer: Total students = 2k + 1 (odd). Since pairing requires 2 students per group, 2k students can be paired, leaving 1 student unpaired, proving the statement.

Question 1: Prove that the product of two odd numbers is always odd.

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

Question 2: Using proof by contradiction, prove that there is no even prime number greater than 2.

Answer: Assume an even prime p > 2. Then p = 2k, with k > 1. Since p is prime, p has no divisors other than 1 and p, but p = 2k, and k > 1 implies p divisible by 2 and k, contradicting primality. Hence, the only even prime is 2.

Question 1: Is the number 2 an odd or even prime? Justify your answer.

Answer: 2 is even and prime because it has only two divisors: 1 and itself.

Question 2: Show that the sum of two odd numbers can be even or odd? (Explain with examples.)

Answer: Sum of two odd numbers is always even: (3 + 5) = 8. Sum of an odd and an even number is odd: (3 + 4) = 7.

Question 1: Statement: All prime numbers are odd. Correct the mistake and explain.

Answer: The statement is false because 2 is a prime number that is even. Correct statement: All prime numbers except 2 are odd.

Question 2: Statement: The product of two even numbers is always odd. Explain and correct.

Answer: The statement is false. The product of two even numbers is always even, e.g., 4 × 6 = 24. Correct statement: The product of two even numbers is always even.