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
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Correct Answer: 6272
Solution : Let the two numbers be a and b. Given, $a^3+b^3=9728$ and $a+b=32$ Now, $a^3+b^3=9728$ ⇒ $(a+b)^3–3ab(a+b)=9728$ ⇒ $32^3–3ab(32)=9728$ ⇒ $96ab=32768–9728$ ⇒ $ab=240$ ⇒ $(a–b)^2=(a+b)^2–4ab=32^2–4(240)=64$ ⇒ $a–b=8$ So, $a^3–b^3=(a–b)^3+3ab(a–b)=8^3+3(240)(8)=512+5760=6272$ Hence, the correct answer is 6272.
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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 sum of the cubes of the two given numbers?
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?
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?
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.
Question : The cube of the sum of two given numbers is 1728, while the product of the two given numbers is 32. Find the positive difference between the cubes of the two given numbers.
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