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Find the number of pairs of positive integers (x, y) su

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by varun289 » Sun Dec 16, 2012 5:19 am
Find the number of pairs of positive integers (x, y) such that x^6 = y^2 + 127.
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
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by Anurag@Gurome » Sun Dec 16, 2012 6:30 am
varun289 wrote:Find the number of pairs of positive integers (x, y) such that x^6 = y^2 + 127
--> (x^6 - y^2) = 127
--> (x^3 - y)(x^3 + y) = 127

As 127 is a prime number, it can be expressed as the product of two integers in only one way, i.e. 1*127

Hence, either
  • (x^3 - y) = 1 and (x^3 + y) = 127
    --> x = 4 and y = 63
OR
  • (x^3 - y) = 127 and (x^3 + y) = 1
    --> x = 4 and y = -63 ---> Not possible
Hence, there is only one such pair of x and y.

The correct answer is B.
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