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If p, q, x, and y are non-zero integers and

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gmatophobia
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If p, q, x, and y are non-zero integers and [#permalink] If p, q, x, and y are non-zero integers and   03 Dec 2023, 10:43
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If \(p, q, x,\) and \(y\) are non-zero integers and '\(x\)' is even, is \(p * (x^q * y)\) = even

1) '\(q\)' is an odd integer

2) '\(p\)' is an even integer
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VivekPrateek
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If p, q, x, and y are non-zero integers and [#permalink] If p, q, x, and y are non-zero integers and   07 Dec 2023, 22:59
gmatophobia wrote:
If \(p, q, x,\) and \(y\) are non-zero integers and '\(x\)' is even, is \(p * (x^q * y)\) = even

1) '\(q\)' is an odd integer

2) '\(p\)' is an even integer


In my take statement 1 and Statement 2 are not necessary to give the answer as the answer is always even no matter whether q/p is even or odd.

Why?

x =even so all the power of x is always even.

when even is multiplied with any number it will always be even, so what the value of q is not necessary.

similarly the (x^q * y) is even so any nonzero number when multiplied with it will always be even so value of p is not necessary.

So OA is E.
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If p, q, x, and y are non-zero integers and [#permalink] If p, q, x, and y are non-zero integers and   08 Dec 2023, 00:31
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VivekPrateek wrote:
gmatophobia wrote:
If \(p, q, x,\) and \(y\) are non-zero integers and '\(x\)' is even, is \(p * (x^q * y)\) = even

1) '\(q\)' is an odd integer

2) '\(p\)' is an even integer


In my take statement 1 and Statement 2 are not necessary to give the answer as the answer is always even no matter whether q/p is even or odd.

Why?

x =even so all the power of x is always even.

when even is multiplied with any number it will always be even, so what the value of q is not necessary.

similarly the (x^q * y) is even so any nonzero number when multiplied with it will always be even so value of p is not necessary.

So OA is E.


VivekPrateek - I believe the highlighted portions of the statements are not very accurate.

Consider this

If x is even, \(x^q \) is NOT guaranteed to be even for all integer values of \(q\). The obvious example is when the power is zero. \(x^0 = 1\), which is not even.

Another scenario is when the power is negative.

Example : \(x = 2, x^{-1} = \frac{1}{2}\).

\(\frac{1}{2}\) is not even.

The answer to the above question is not E, because the value is always even regardless of the statements. If that were the case, the answer would have been D (and the question would be an incorrect DS question to begin with :lol: )
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Re: If p, q, x, and y are non-zero integers and [#permalink] If p, q, x, and y are non-zero integers and   08 Dec 2023, 00:44
gmatophobia wrote:
If \(p, q, x,\) and \(y\) are non-zero integers and '\(x\)' is even, is \(p * (x^q * y)\) = even

1) '\(q\)' is an odd integer

2) '\(p\)' is an even integer


Official Explanation

Given

  • \(p, q, x,\) and \(y\) are non-zero integers
  • \(x\) is even

Question

Is \(p * (x^q * y)\) = even

Statement 1

1) '\(q\)' is an odd integer

While we know that the value of q is odd, we do not know whether the value is positive or negative. Without that information, we cannot determine the even/odd nature of \(x^q\)

Consider the following -

Case 1

\(x = 2, p = 2 , q = -1, y = 1\)

\(2 * (2^{-1} * 1) = 2 * (\frac{1}{2} * 1) = 1\)

The value is odd.

Case 2

\(x = 2, p = 2 , q = 1, y = 1\)

\(2 * (2^{1} * 1) = 2 * (2 * 1) = 4\)

The value is even.

As we have multiple possible answers, the statement alone is not sufficient and we can eliminate A and D.

Statement 2

2) '\(p\)' is an even integer

We know that \(p\) is even, however, we do not know the positive-negative nature of \(q\). The cases taken to evaluate Statement 1 are valid in this case as well.

We know that Statement 1 is not sufficient, hence Statement 2 is also not sufficient to arrive at a definite answer.

Eliminate B.

Combined

Statement 2 doesn't provide any additional information to resolve the positive negative nature of q. Hence, the statements combined are not sufficient as well.

Option E
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Re: If p, q, x, and y are non-zero integers and [#permalink] If p, q, x, and y are non-zero integers and   08 Dec 2023, 01:24
gmatophobia wrote:
VivekPrateek wrote:
gmatophobia wrote:
If \(p, q, x,\) and \(y\) are non-zero integers and '\(x\)' is even, is \(p * (x^q * y)\) = even

1) '\(q\)' is an odd integer

2) '\(p\)' is an even integer


In my take statement 1 and Statement 2 are not necessary to give the answer as the answer is always even no matter whether q/p is even or odd.

Why?

x =even so all the power of x is always even.

when even is multiplied with any number it will always be even, so what the value of q is not necessary.

similarly the (x^q * y) is even so any nonzero number when multiplied with it will always be even so value of p is not necessary.

So OA is E.


VivekPrateek - I believe the highlighted portions of the statements are not very accurate.

Consider this

If x is even, \(x^q \) is NOT guaranteed to be even for all integer values of \(q\). The obvious example is when the power is zero. \(x^0 = 1\), which is not even.

Another scenario is when the power is negative.

Example : \(x = 2, x^{-1} = \frac{1}{2}\).

\(\frac{1}{2}\) is not even.

The answer to the above question is not E, because the value is always even regardless of the statements. If that were the case, the answer would have been D (and the question would be an incorrect DS question to begin with :lol: )


Thanks for correcting me gmatophobia . Since they are non zero, I missed to take care of negative one. Thanks :)
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Re: If p, q, x, and y are non-zero integers and [#permalink]
08 Dec 2023, 01:24
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