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 sum of the cubes of these two given numbers.
Option 1: 6048
Option 2: 5616
Option 3: 6024
Option 4: 5832
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Correct Answer: 6048
Solution :
Let the two numbers be $a$ and $b$.
$(a–b)^3=1728$ and $ab=108$.
⇒ $(a–b)=\sqrt[3]{1728}$
⇒ $(a–b)=12$ -------------------------------(equation 1)
Now, $(a+b)^2=(a–b)^2+4ab$
⇒ $(a+b)^2=(12)^2+4(108)$
⇒ $(a+b)=\sqrt{144+432}$
⇒ $(a+b)=24$ ---------------------------(equation 2)
Performing equation (2) – equation (1), we get:
$(a+b)-(a-b) = 24-12$
⇒ $2b = 12$
⇒ $b=6$
Substituting it in equation 1, we get:
$a-6= 12$
⇒ $a=18$
Therefore, $a^3+b^3=18^3+6^3=5832+216=6048$.
Hence, the correct answer is 6048.
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