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
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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