Question : The sum of the cubes of two given natural numbers is 9728, while the sum of the two given numbers is 32. What is the positive difference between the cubes of the two given numbers?
Option 1: 6272
Option 2: 5832
Option 3: 4662
Option 4: 7904

Question

Question : The difference between the cubes of two given natural numbers is 6272, while the positive difference between the two given numbers is 8. What is the sum of the cubes of the two given numbers?
Option 1: 9728
Option 2: 9684
Option 3: 8000
Option 4: 9600

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Answer 1

Correct Answer: 6272

Solution : Let the two numbers be a and b.
Given, $a^{3} + b^{3} = 9728$ and $a + b = 32$
Now, $a^{3} + b^{3} = 9728$
⇒ $(a + b)^{3} - 3 a b (a + b) = 9728$
⇒ $32^{3} - 3 a b (32) = 9728$
⇒ $96 a b = 32768 - 9728$
⇒ $a b = 240$
⇒ $(a - b)^{2} = (a + b)^{2} - 4 a b = 32^{2} - 4 (240) = 64$
⇒ $a - b = 8$
So, $a^{3} - b^{3} = (a - b)^{3} + 3 a b (a - b) = 8^{3} + 3 (240) (8) = 512 + 5760 = 6272$
Hence, the correct answer is 6272.

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Question : The sum of the cubes of two given natural numbers is 9728, while the sum of the two given numbers is 32. What is the positive difference between the cubes of the two given numbers?Option 1: 6272Option 2: 5832Option 3: 4662Option 4: 7904

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Question : The sum of the cubes of two given natural numbers is 9728, while the sum of the two given numbers is 32. What is the positive difference between the cubes of the two given numbers?
Option 1: 6272
Option 2: 5832
Option 3: 4662
Option 4: 7904