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If x, y and z are positive integers such that x^4*y^3 = z^2
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Rakesh1987
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Re: If x, y and z are positive integers such that x^4*y^3 = z^2 [#permalink] New post  02 Jun 2019, 23:21
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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.

The answer is D.

Please hit +1 Kudos if you liked the answer. :cool:
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Re: If x, y and z are positive integers such that x^4*y^3 = z^2 [#permalink] New post  10 Aug 2020, 12:26
bloody hell, is it only me can't see the options!?
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Re: If x, y and z are positive integers such that x^4*y^3 = z^2 [#permalink] New post  10 Aug 2020, 21:49
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VIVIANMWW wrote:
bloody hell, is it only me can't see the options!?



This is a data sufficiency question. Options for DS questions are always the same.

The data sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements, plus your knowledge of mathematics and everyday facts (such as the number of days in July or the meaning of the word counterclockwise), you must indicate whether—

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
D. EACH statement ALONE is sufficient to answer the question asked.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

I suggest you to go through the following post ALL YOU NEED FOR QUANT.

Hope this helps.
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Re: If x, y and z are positive integers such that x^4*y^3 = z^2 [#permalink] New post  12 Aug 2020, 01:27
Hi Harsh,

Can you please help me with the following understanding:

Under Step-III: Statement-I, it says "We observe here that a fraction has been simplified to an odd number (as 4k is always even and even + odd = odd) which would imply that both the numerator and the denominator are either even or odd.

Is this 'implication' a conceptual understanding that we ought to know or a logical understanding?
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Re: If x, y and z are positive integers such that x^4*y^3 = z^2 [#permalink] New post  28 Sep 2020, 19:36
Hi, this is honestly a very difficult question to understand. When I was going in practice mode, I didn't get it correct. I read the solutions and I guess it was just the words that made it difficult for me to understand the solution. So, I have recreated the solutions already provided by multiple instructors and members. I have added some of my understanding to the solution.
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If x, y and z are positive integers such that x^4*y^3 = z^2 [#permalink] New post  07 Mar 2021, 13:28
Harley1980 wrote:
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) \(z = x + y\)

We will provide the OA in some time. Till then Happy Solving :lol:

This is

Ques 10 of The E-GMAT Number Properties Knockout



Register for our Free Session on Number Properties this Saturday to solve exciting 700+ Level Questions in a classroom environment under the real-time guidance of our Experts! :)


In this task we know that x, y, z are postive integers so we can eliminate all powers from all equations because they don't have influence on parity
1) we know that \(\frac{xy}{(x+y)}\) equal to odd result (4k + 3). This is possible only if xy and x+y have the same parity.
And they can have the same parity only if they both even.
So if they both even when we can say that z is too even
Sufficient

2) from this statement we know that z = x + y and from task we know that z = xy
This is possible only if xy, (x+y) and z have the same parity.
And they can have the same parity only if they even.
So z is even
Sufficient

Answer is D


I get the process behind this but am confused about statement 1.

4k + 3 = odd so how can an even / even = even be right? Can't it be odd / odd? e.g. 9/3 = 3 (odd).

OK nvm. The question makes total sense now.

First, we have to deduce from x^4*y^3 = z^2 that x and y are both odd or both even.
Statement 1-
Case 1: Suppose they are both odd, then the numerator becomes odd and the denominator is even ...so we end up with a non-integer
Case 2: Suppose they are both even, then you get an integer and the answer to the question is YES ...even.

Sufficient

Statement 2:
x + y and zy can only have the same even/odd nature if x and y are each even
Sufficient

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