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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


Team Careers360 20th Jan, 2024
Answer (1)
Team Careers360 23rd Jan, 2024

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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