Suppose m = HCF of x and y and n = LCM of x and y. Consider the following :
1. m²n ≤x²y
2. mn² ≤x²y
Which of the following is/are correct ?

The relationship between two numbers (x, y) and their HCF (m) and LCM (n) is defined by the fundamental property: x × y = m × n.

• Statement 1: m²n ≤ x²y. We can rewrite this as m(mn) ≤ x(xy). Substituting mn = xy, we get m(xy) ≤ x(xy). Dividing both sides by xy (assuming x, y > 0), we get m ≤ x. Since the HCF of two numbers is always less than or equal to the numbers themselves, this statement is correct.
• Statement 2: mn² ≤ x²y. We can rewrite this as (mn)n ≤ x(xy). Substituting mn = xy, we get (xy)n ≤ x(xy), which simplifies to n ≤ x. This is incorrect because the LCM (n) is always greater than or equal to the numbers (n ≥ x and n ≥ y).

Thus, only statement 1 is correct.