State whether the following statements are true or false. If a statement is false, justify your answer:

(i) The sum of two prime numbers is always an even number.

(ii) The sum of two prime numbers is always a prime number.

(iii) The sum of two prime numbers can never be a prime number.

(iv) No odd number can be written as the sum of two prime numbers.

(v) If two numbers are co-prime, then atleast one of them must be prime.

(vi) If a number is divisible by 18, it must be divisible by 3 and 6 both.

(vii) If a number is divisible by 2 and 4 both, it must be divisible by 8.

(viii) If a number is divisible by 3 and 6 both, it must be divisible by 18.

(ix) HCF of an even number and an odd number is always 1.

(i) False.

Reason: For example, 2 and 7 both are prime numbers but their sum = 2 + 7 = 9, which is an odd number.

(ii) False.

Reason: For example, 3 and 5 both are prime numbers but their sum = 3 + 5 = 8, which is a composite number.

(iii) False.

Reason: For example, 2 and 5 both are prime numbers but their sum = 2 + 5 = 7, which is a prime number.

(iv) False.

Reason: For example,13 is an odd number and 13 = 2 + 11, which is the sum of two prime numbers.

(v) False.

Reason: For example, 8 and 15 are co-prime but neither 8 is prime nor 15 is prime.

(vi) True.

Reason: Since 3 and 6 are factors of 18, any number divisible by 18 is divisible by both 3 and 6.

(vii) False.

Reason: For example, 20 is divisible by 2 and 4 both but not divisible by 8.

(viii) False.

Reason: 12 is divisible by 3 and 6 both but 12 is not divisible by 18.

(ix) False.

Reason: For example, 6 is even and 9 is odd but HCF(6, 9) = 3.