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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?

Option 1: 9728

Option 2: 9684

Option 3: 8000

Option 4: 9600


Team Careers360 8th Jan, 2024
Answer (1)
Team Careers360 17th Jan, 2024

Correct Answer: 9728


Solution : Given: $a^{3}–b^{3}=6272$ and $a–b=8$
We know that $(a–b)^{3}=a^{3}–b^{3}–3ab(a–b)$
Putting the given value, we have,
⇒ $(8)^{3}=6272–3ab(8)$
⇒ $24ab=6272–512$
⇒ $ab=\frac{5760}{24}=240$ ------------------------------------(1)
Putting the value of $a=b+8$ we have,
⇒ $(b+8)b=240$
⇒ $b^{2}+8b–240=0$
⇒ $b^{2}+20b–12b–240=0$
⇒ $b(b+20)–12(b+20)=0$
⇒ $(b+20)(b–12)=0$
⇒ $b=12,–20$
The value of a natural number can't be negative.
Putting the value of $b=12$ we have,
$a=20$
Now, we find, $a^{3}+b^{3}=20^{3}+12^{3}$
⇒ $a^{3}+b^{3}=8000+1728$
⇒ $a^{3}+b^{3}=9728$
Hence, the correct answer is 9728.

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