If the product of the HCF and LCM equals the cube of one number, it suggests that the relation $HCF \times LCM = a^3$ for distinct numbers $a$ and $b$ implies a specific form in which $b$ shares prime factors with $a$. This often results in $a$ being a perfect square (to ensure integer results for $b$), and the difference between perfect squares is always even, validating both statements.