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
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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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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?
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 positive difference between 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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