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integers!

by Ozlemg » Mon Jul 04, 2011 1:05 pm
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 Frankenstein » Mon Jul 04, 2011 6:59 pm
Hi,
127 is a prime. It can be written as product of two factors in only one way i.e. 1*127
x^6-y^2 = 127
So, (x^3+y)(x^3-y) = 127 = 127*1
So, x^3+y = 127, x^3-y = 1
So, x^3 = 64 => x=4
and y =63.
Hence, B
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