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Algebra Level pending

find the number of all positive integral pairs (x,y)

( x y − 7 ) 2 = x 2 + y 2 (xy - 7)^{2} = x^{2} + y^{2}

please post your solution


The answer is 4.

U Z by U Z , 24, Guyana

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1 solution

U Z
Oct 12, 2014

( x y ) 2 − 14 x y + 49 = x 2 + y 2 (xy)^{2} - 14xy + 49 = x^{2} +y^{2}

= ( x y ) 2 − 2.6 x y + 36 + 13 = x 2 + y 2 = (xy)^{2} - 2.6xy + 36 + 13 = x^{2} +y^{2} + 2xy)

= ( x y − 6 ) 2 + 13 = ( x + y ) 2 = (xy - 6)^{2} + 13 = (x+ y)^{2}

\(13 = [(x+y + xy - 6)(x + y -xy +6)

since 13 is a prime number therefore

\(x + y + xy - 6 = \pm 13\)

or

x + y − x y + 6 = ± 1 x+ y - xy +6 = \pm 1

thus (x, y) = (7 , 0) , (0 , 7) , (3 ,4) and (4 , 3)

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