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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


Team Careers360 21st Jan, 2024
Answer (1)
Team Careers360 23rd Jan, 2024

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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