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?
Show answer & explanation
Correct answer: A
Explanation
The correct answer is option 1 i.e. 6272.
Let the two numbers be x and y.
⇒ x3 + y3 = 9728
⇒ x + y = 32
x3 + y3 = (x + y)3 - 3xy(x + y)
⇒ 9728 = (32)3 - 3xy(32)
⇒ 9728 - 32768 = -96xy
xy = 23040/96 = 240
(x - y)2 = (x + y)2 - 4xy
⇒ (x - y)2 = (32)2 - 960
⇒ (x - y) = 8
x3 − y3 = (x - y)3 + 3xy(x - y) = 83 + 3 × 240(8)
= 512 + 5760
= 6272