If the product of the HCF and LCM of two distinct numbers is the cube of one of the numbers, then which of the following statements is/are correct?
I. The difference of the numbers is an even number.
II. One of the numbers is a perfect square.

Select the answer using the code given below.

Correct Option: (d) Neither I nor II

Let the two distinct numbers be a and b. We know that for any two numbers, HCF(a,b) × LCM(a,b) = a × b. According to the question, HCF × LCM = a^3 (assuming one of the numbers is a), so a × b = a^3, which implies b = a^2. Since a and b are distinct positive integers, if b = a^2, then one of the numbers (b) is indeed a perfect square (Statement II is correct). Let's test Statement I: The difference of the numbers is a - b = a - a^2 = a(1 - a). If a = 2, b = 4, the difference is 4 - 2 = 2 (even). If a = 3, b = 9, the difference is 9 - 3 = 6 (even). If a = 1, b = 1 (not distinct). For any integer a >= 2, a^2 - a = a(a - 1), which is always the product of two consecutive integers and hence always an even number (Statement I is correct). Therefore, both statements I and II are correct.