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

Question

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.

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Answer 1

Correct Answer: 6048

Solution : Let the two numbers be $a$ and $b$ .
$(a - b)^{3} = 1728$ and $a b = 108$ .
⇒ $(a - b) = \sqrt[3]{1728}$
⇒ $(a - b) = 12$ -------------------------------(equation 1)
Now, $(a + b)^{2} = (a - b)^{2} + 4 a b$
⇒ $(a + b)^{2} = (12)^{2} + 4 (108)$
⇒ $(a + b) = \sqrt{144 + 432}$
⇒ $(a + b) = 24$ ---------------------------(equation 2)
Performing equation (2) – equation (1), we get:
$(a + b) - (a - b) = 24 - 12$
⇒ $2 b = 12$
⇒ $b = 6$
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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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: 6048Option 2: 5616Option 3: 6024Option 4: 5832

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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