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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