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 product of the two given numbers?
Option 1: 160
Option 2: 320
Option 3: 240
Option 4: 200
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Correct Answer: 240
Solution : Let the two numbers be $a$ and $b$, According to the question, $a + b = 32$ and, $a^{3}$ + $b^{3} = 9728$ Now, $(a+b)^{3} = a^{3}+b^{3}+3ab(a+b)$ ⇒ $32^{3} = 9728$ + $3ab(32)$ ⇒ $ab=\frac{32768 − 9728}{96}$ $\therefore$ $ab = 240$ Hence, the correct answer is 240.
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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 product of the two given numbers?
Option 2: 240
Option 3: 200
Option 4: 320
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 : 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
Question : The cube of the sum of two given numbers is 1728, while the product of the two given numbers is 32. Find the sum of the cubes of the two given numbers.
Option 1: 576
Option 2: 250
Option 3: 640
Option 4: 512
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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