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.

According to Euclidean mathematics and the Fundamental Theorem of Arithmetic (dating to Euclid's Elements, c. 300 BC), a core property of two positive integers a and b is that the product of their Highest Common Factor (HCF) and Least Common Multiple (LCM) equals the product of the numbers themselves: HCF × LCM = a × b.

The question states this product equals the cube of one number. Let this be a³. Thus, a × b = a³, which simplifies to b = a². Because the numbers are distinct, a ≠ 1.

Statement I (The difference of the numbers is an even number): The difference between the numbers is b - a = a² - a = a(a - 1). In number theory, a and a-1 are consecutive integers. Because one of any two consecutive integers must be even, their product is always even. Thus, Statement I is correct.

Statement II (One of the numbers is a perfect square): We algebraically derived that b = a². This inherently makes b a perfect square. Thus, Statement II is also correct.

Because both mathematically derived statements hold true, Option C is the correct answer.

Why other options are wrong:

• Option A (I only): Incorrect because it fails to acknowledge that b = a² explicitly guarantees Statement II.
• Option B (II only): Incorrect because it ignores the consecutive integer parity rule that guarantees an even difference, validating Statement I.
• Option D (Neither I nor II): Incorrect because both statements are logically and mathematically proven to be true.

Takeaway: Mnemonic: "Product equals Product" — HCF × LCM = N₁ × N₂. Additionally, remember that the product of any two consecutive integers, modeled as n(n-1), is always an even number.