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# If x, y and z are positive integers such that x^4*y^3 = z^2

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Joined: 17 Jul 2018
Posts: 67
GMAT 1: 760 Q50 V44
Re: If x, y and z are positive integers such that x^4*y^3 = z^2  [#permalink]

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03 Jun 2019, 00:21
EgmatQuantExpert wrote:
If $$x, y$$ and $$z$$ are positive integers such that $$x^4y^3 = z^2$$, is $$x^9 - y^6$$ odd?

(1) $$\frac{x^4y^3}{(x^2 + y^2)}$$ can be written in the form $$4k + 3$$, where $$k$$ is a positive integer

(2) $$z = x + y$$

Statement 1 gives that $$\frac{x^4y^3}{(x^2 + y^2)}$$ is odd integer
Case 1) X= odd, y= odd, makes denominator $$(x^2 + y^2)$$ = O+O=E and numerator is odd which makes the expression Not an integer. Not possible
Case 2) X= Even, y= Even, makes denominator $$(x^2 + y^2)$$ = E+E=E and the numerator a multiple of two even numbers and hence even. The multiples of 2 in numerator and denominator can cancel out to make the simplified expression an odd integer. Possible. Here $$x^9 - y^6$$= E-E =E.
Case 3) one of x,y is E and other O, makes denominator $$(x^2 + y^2)$$ = E+O= O, the numerator is even as it is a multiple of even number, then it cant be odd. Not possible
Statement 1 is sufficient.

Statement 2 gives $$z = x + y$$ and we already have $$x^4y^3 = z^2$$.
Case 1) X= O, y= O, gives Z =E, and $$x^4y^3 = z^2$$ gives Z=O*O=O. Not possible.
Case 2) X= E, y= E, gives Z =E, and $$x^4y^3 = z^2$$ gives Z=E*E=E. Possible. Here $$x^9 - y^6$$= E-E =E.
Case 3) one of x,y is E and other O gives Z=O+E=O, and $$x^4y^3 = z^2$$ gives Z=E*O=E. Not possible.
Statement 2 is sufficient.

Re: If x, y and z are positive integers such that x^4*y^3 = z^2   [#permalink] 03 Jun 2019, 00:21

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