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Math Expert V
Joined: 02 Sep 2009
Posts: 55802

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Difficulty:   75% (hard)

Question Stats: 46% (01:32) correct 54% (01:00) wrong based on 116 sessions

HideShow timer Statistics If $$x$$ and $$y$$ are even integers, which of the following must also be an even integer?

A. $$x^y$$
B. $$2\frac{x}{y}$$
C. $$\frac{x - y}{x + y}$$
D. $$\frac{x^2 - y^2}{2}$$
E. $$(x + 1)(y - 1)$$

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Math Expert V
Joined: 02 Sep 2009
Posts: 55802

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Official Solution:

If $$x$$ and $$y$$ are even integers, which of the following must also be an even integer?

A. $$x^y$$
B. $$2\frac{x}{y}$$
C. $$\frac{x - y}{x + y}$$
D. $$\frac{x^2 - y^2}{2}$$
E. $$(x + 1)(y - 1)$$

$$\frac{x^2 - y^2}{2} = \frac{(x - y)(x + y)}{2} = \frac{even*even}{2} = even*integer = even$$.

$$x^y$$ is 1 if $$y = 0$$ and $$x$$ is positive.

$$2*\frac{x}{y}$$ is 1 if $$x = 2$$ and $$y = 4$$.

$$\frac{x - y}{x + y}$$ might not be an integer at all.

$$(x + 1)(y - 1)$$ is always odd.

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Here, if x=y (Question didn't say that x and y are different, so we can consider this use case as well), then option (D) becomes 0/2 =0, which is not an even integer. Can you please let me know your thoughts on this?
Math Expert V
Joined: 02 Sep 2009
Posts: 55802

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reddyMBA wrote:
Here, if x=y (Question didn't say that x and y are different, so we can consider this use case as well), then option (D) becomes 0/2 =0, which is not an even integer. Can you please let me know your thoughts on this?

You should brush up fundamentals before attempting questions.

ZERO:

1. 0 is an integer.

2. 0 is an even integer. An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without a remainder and as zero is evenly divisible by 2 then it must be even.

3. 0 is neither positive nor negative integer (the only one of this kind).

4. 0 is divisible by EVERY integer except 0 itself.

Check more here: tips-and-hints-for-specific-quant-topics-with-examples-172096.html#p1371030
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Intern  B
Joined: 24 Jul 2017
Posts: 37
GMAT 1: 710 Q48 V40 WE: Research (Energy and Utilities)

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Option 2 in the question looks like a Mixed fraction and not 2*(x/y)
Manager  B
Joined: 19 Oct 2017
Posts: 56
Location: India
GMAT 1: 710 Q50 V35 GPA: 3.6
WE: Analyst (Commercial Banking)

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Bunuel wrote:
Official Solution:

If $$x$$ and $$y$$ are even integers, which of the following must also be an even integer?

A. $$x^y$$
B. $$2\frac{x}{y}$$
C. $$\frac{x - y}{x + y}$$
D. $$\frac{x^2 - y^2}{2}$$
E. $$(x + 1)(y - 1)$$

$$\frac{x^2 - y^2}{2} = \frac{(x - y)(x + y)}{2} = \frac{even*even}{2} = even*integer = even$$.

$$x^y$$ is 1 if $$y = 0$$ and $$x$$ is positive.

$$2*\frac{x}{y}$$ is 1 if $$x = 2$$ and $$y = 4$$.

$$\frac{x - y}{x + y}$$ might not be an integer at all.

$$(x + 1)(y - 1)$$ is always odd.

Hello. How can I know the difficulty level of this or any other quant question (part of gmat club test)?

Sent from my Redmi Note 4 using GMAT Club Forum mobile app
Math Expert V
Joined: 02 Sep 2009
Posts: 55802

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HarshBazooka wrote:
Bunuel wrote:
Official Solution:

If $$x$$ and $$y$$ are even integers, which of the following must also be an even integer?

A. $$x^y$$
B. $$2\frac{x}{y}$$
C. $$\frac{x - y}{x + y}$$
D. $$\frac{x^2 - y^2}{2}$$
E. $$(x + 1)(y - 1)$$

$$\frac{x^2 - y^2}{2} = \frac{(x - y)(x + y)}{2} = \frac{even*even}{2} = even*integer = even$$.

$$x^y$$ is 1 if $$y = 0$$ and $$x$$ is positive.

$$2*\frac{x}{y}$$ is 1 if $$x = 2$$ and $$y = 4$$.

$$\frac{x - y}{x + y}$$ might not be an integer at all.

$$(x + 1)(y - 1)$$ is always odd.

Hello. How can I know the difficulty level of this or any other quant question (part of gmat club test)?

Sent from my Redmi Note 4 using GMAT Club Forum mobile app

You can check the stats in the original post:
Attachment:
2018-02-07_1755.png

>> !!!

You do not have the required permissions to view the files attached to this post.

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I used test cases and got down to two answer choices: A and D. I couldn't find an example of two even integers that disproved A?

Math Expert V
Joined: 02 Sep 2009
Posts: 55802

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DouglassJensen wrote:
I used test cases and got down to two answer choices: A and D. I couldn't find an example of two even integers that disproved A?

Consider x = 2 and y = 0 --> 2^0 = 1 = odd.

(non-zero integer)^0 = 1 = odd.
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Senior Manager   G
Joined: 31 May 2017
Posts: 339

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After initial analysis , answer choices narrowed down to A and D. The option "0" was the caveat in the choice A which made the only possible option is D.

Thanks Bunuel for clear explanation as always
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Intern  B
Joined: 30 Aug 2017
Posts: 10
GMAT 1: 580 Q45 V26 GMAT 2: 700 Q48 V38 GMAT 3: 710 Q49 V39 Show Tags

I think this is a high-quality question and I agree with explanation.
Intern  B
Joined: 22 Mar 2017
Posts: 4

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Hi Bunuel,

Using the same logic( i.e. 0 is neither even nor odd) shouldn't A also satisfy the conditions?

x^y would always be even as y can never be 0(as 0 is not an even number).

Thanks,
Sam
Math Expert V
Joined: 02 Sep 2009
Posts: 55802

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samsam00 wrote:
Hi Bunuel,

Using the same logic( i.e. 0 is neither even nor odd) shouldn't A also satisfy the conditions?

x^y would always be even as y can never be 0(as 0 is not an even number).

Thanks,
Sam

0 is definitely an even integer. Have you checked this post above: https://gmatclub.com/forum/m22-184324.html#p1517172
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Intern  B
Joined: 22 Mar 2017
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Thanks, Bunuel. That was rather a stupid query from my end.
Intern  G
Joined: 26 Jul 2017
Posts: 48
Location: India
GMAT 1: 570 Q48 V20 WE: Information Technology (Computer Software)

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I think this is a high-quality question and I agree with explanation.
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