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 sum of the cubes of these two given numbers.
Option 1: 6048
Option 2: 5616
Option 3: 6024
Option 4: 5832
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Correct Answer: 6048
Solution : Let the two numbers be $a$ and $b$. $(a–b)^3=1728$ and $ab=108$. ⇒ $(a–b)=\sqrt[3]{1728}$ ⇒ $(a–b)=12$ -------------------------------(equation 1) Now, $(a+b)^2=(a–b)^2+4ab$ ⇒ $(a+b)^2=(12)^2+4(108)$ ⇒ $(a+b)=\sqrt{144+432}$ ⇒ $(a+b)=24$ ---------------------------(equation 2) Performing equation (2) – equation (1), we get: $(a+b)-(a-b) = 24-12$ ⇒ $2b = 12$ ⇒ $b=6$ Substituting it in equation 1, we get: $a-6= 12$ ⇒ $a=18$ Therefore, $a^3+b^3=18^3+6^3=5832+216=6048$. Hence, the correct answer is 6048.
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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.
Option 1: 4104
Option 3: 2160
Option 4: 5626
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.
Option 1: 448
Option 2: 576
Option 3: 480
Option 4: 512
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
Question : The cube of the sum of two given numbers is 1728, while the product of the two given numbers is 32. Find the sum of the cubes of the two given numbers.
Option 1: 576
Option 2: 250
Option 3: 640
Question : The square of the difference between two given natural numbers is 324, while the product of these two given numbers is 144. Find the positive difference between the squares of these two given numbers.
Option 1: 630
Option 2: 540
Option 3: 450
Option 4: 360
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