Find the number of ordered pairs (x, y) of positive integers such that (x + y)(x2 + 9y) is the cube of a prime number.
Answers
Given : (x+y)(x² +9y) is the cube of a prime
To Find : number of ordered pairs (x, y) of positive integers
Solution:
(x+y)(x² +9y) is the cube of a prime
Hence
x² +9y = (x + y)² as x² ≥ x and 9y > y => x² +9y > x + y
and neither of x + y and x² +9y can be 1
=> x² + 9y = x² + y² + 2xy
=> y² + 2xy = 9y
=> y + 2x = 9
2x + y = 9
=> x = 1 , y = 7
x = 2 , y = 5
x = 3 , y = 3
x = 4 , y = 1
Ordered pairs ( 1 , 7) , ( 2 , 5) , ( 3 , 3 ) , ( 4 ,1 )
4 ordered pairs
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