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?

Correct option is A

Given:
Sum of the cubes of two numbers = 9728
a³ + b³ = 9728
Sum of two numbers = 32
a + b = 32
Formula Used:
a³ + b³ = (a + b) (a² + b² – ab)
a³ – b³ = (a - b) (a² + b² + ab)
Solution:
Let two numbers are a and b.
a³ + b³ = (a + b) [(a + b)² – 3ab]
9728 = 32[(32)² -3×ab]
304 = [1024 – 3ab]
3ab = 720
ab = 240
(a – b)² = (a+b)² – 4ab
(a – b)² = (32)² - 4×240
(a – b)² = 1024 – 960
(a-b )² = 64
(a-b) = 8
Then the value of a³ – b³ = (a - b) (a² + b² + ab)
= (a - b) ((a - b )² +3 ab)
= 8×[(8)² +3×240]
= 8×[64+ 720]
= 8×784
= 6272
Alternate Method:
a + b = 32
a³ + b³ = 9728
by solving both equation
ab = 240 = 20×12
so, a = 20, b = 12
find a³ - b³ = 8000 - 1728
= 6272