Question : The cube of the difference between two given natural numbers is 1728, while the product of these two given numbers is 108. Find the positive difference between the cubes of these two given numbers.
Option 1: 4104
Option 2: 5616
Option 3: 2160
Option 4: 5626
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Correct Answer: 5616
Solution : Given: The cube of the difference between two given natural numbers is 1728, while the product of these two given numbers is 108. Let the two numbers be $a$ and $b$. So, $(a–b)^3=1728$ and $ab=108$. Now, $a^3–b^3=(a–b)^3+3ab(a–b)$ ⇒ $a^3–b^3=1728+3×108(12)$ ⇒ $a^3–b^3=1728+3888$ ⇒ $a^3–b^3=5616$ Hence, the correct answer is 5616.
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