Question
Download Solution PDFThe 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?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDF
x3 - y3 = 6272
x - y = 8
Formula used:-
( A3 + B3) = ( A + B)3 - 3AB( A + B)
( A3 - B3) = ( A - B)3 + 3AB( A - B)
Calculation:-
According to the formula,
x3 - y3 = ( x - y )3 + 3xy(x - y)
⇒ ( x - y )3 + 3xy(x - y) = 6272
⇒ 83 + 3xy(8) = 6272
⇒24xy = 6272 - 512 = 5760
⇒ xy = 240 .....(1)
Now,
(x - y) = 8 [Given]
Square both sides,
(x - y)2 = 64
⇒ x2 + y2 - 2xy = 64
⇒ x2 + y2 = 64 + 2× 240 [ xy = 240]
⇒ x2 + y2 = 544 ....(2)
We know that
(x + y)2 = x2 + y2 + 2xy
⇒ (x + y)2 = 544 + 2× 240
⇒ (x + y)2 = 1024
⇒ (x + y) = 32
As per the question,
x3 + y3 = ( x + y )3 - 3xy(x + y)
⇒ x3 + y3 = 323 - 3 × 32 × 240
⇒ 32768 - 23040 = 9728
∴ The required answer is 9728.
Last updated on Jul 7, 2025
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