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# If x is a positive integer, which of the following CANNOT be

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If x is a positive integer, which of the following CANNOT be [#permalink]  11 Feb 2013, 10:09
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If x is a positive integer, which of the following CANNOT be expressed as n^2, where n is an integer?

A. x^5
B. x^2 − 1
C. \sqrt{x^8}
D. x^2 + 1
E. \sqrt{x^5}
[Reveal] Spoiler: OA

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Last edited by Bunuel on 11 Feb 2013, 14:23, edited 1 time in total.
Edited the question.
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Re: If x is a positive integer, which of the following CANNOT be [#permalink]  11 Feb 2013, 14:13
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emmak wrote:
If x is a positive integer, which of the following CANNOT be expressed as n^2, where n is an integer?

A. x^5
B. x^2 − 1
C. \sqrt{x^8}
D. x^2 + 1
E. \sqrt{x^5}

The question basically asks: if x is a positive integer, which of the following CANNOT be a perfect square.

Now, if x=1, then options A, B, C and E ARE perfect squares, therefore by POE the correct answer must be D.

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Re: If x is a positive integer, which of the following CANNOT be [#permalink]  12 Feb 2014, 15:10
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Re: If x is a positive integer, which of the following CANNOT be [#permalink]  05 Mar 2014, 23:06
If we take x = 2 & calculate, we cant get the answers;
seems that x has to be taken 1 to execute all the options
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Re: If x is a positive integer, which of the following CANNOT be [#permalink]  27 Mar 2014, 16:12
emmak wrote:
If x is a positive integer, which of the following CANNOT be expressed as n^2, where n is an integer?

A. x^5
B. x^2 − 1
C. \sqrt{x^8}
D. x^2 + 1
E. \sqrt{x^5}

So I guess zero does count as a perfect square then

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J
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Re: If x is a positive integer, which of the following CANNOT be [#permalink]  27 Mar 2014, 20:21
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PareshGmat wrote:
If we take x = 2 & calculate, we cant get the answers;
seems that x has to be taken 1 to execute all the options

Any random value of x will not help you get the answer. Even if you do not try x = 1, you can use reasoning to solve this question.

A. x^5
If x is a number with an even power, such as x = a^4 (a is an integer), then x^5 = a^{20} = n^2
n will be a^{10}, an integer here.

B. x^2 - 1
x^2 - 1 = n^2
You need two consecutive perfect squares. Only 0 and 1 are consecutive perfect squares. Thereafter, the distance between perfect squares keeps increasing. x needs to be a positive integers so if x = 1, n = 0 (an integer)

C. \sqrt{x^8}
\sqrt{x^8} = x^4 = n^2
n will be x^2, an integer here.

D. x^2 + 1
x is a positive integer so it must be at least 1. After 1, there are no two consecutive integers. n cannot be an integer.

E. \sqrt{x^5}
If x is a number with an even power which is a multiple of 4, such as x = a^4 (a is an integer), then \sqrt{x^5} = \sqrt{a^{20}} = a^{10} = n^2
n will be a^5, an integer here.

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Karishma
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Save $100 on Veritas Prep GMAT Courses And Admissions Consulting Enroll now. Pay later. Take advantage of Veritas Prep's flexible payment plan options. Veritas Prep Reviews Manager Joined: 14 Jan 2013 Posts: 158 Concentration: Strategy, Technology GMAT Date: 08-01-2013 GPA: 3.7 WE: Consulting (Consulting) Followers: 2 Kudos [?]: 45 [0], given: 29 Re: If x is a positive integer, which of the following CANNOT be [#permalink] 29 Mar 2014, 01:38 Even though Bunuel and karishma have answered this question, I am not able to digest any fundamental of this .... Not sure how to crack this one... as I got this question in my Veritas prep exam today... _________________ "Where are my Kudos" ............ Good Question = kudos "Start enjoying all phases" & all Sections __________________________________________________________________ http://gmatclub.com/forum/collection-of-articles-on-critical-reasoning-159959.html percentages-700-800-level-questions-130588.html 700-to-800-level-quant-question-with-detail-soluition-143321.html Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 4668 Location: Pune, India Followers: 1073 Kudos [?]: 4781 [2] , given: 163 Re: If x is a positive integer, which of the following CANNOT be [#permalink] 30 Mar 2014, 18:01 2 This post received KUDOS Expert's post Mountain14 wrote: Even though Bunuel and karishma have answered this question, I am not able to digest any fundamental of this .... Not sure how to crack this one... as I got this question in my Veritas prep exam today... Look, the question simply asks which option CANNOT be a perfect square. In the options, x is a positive integer. Can x^5 be a perfect square? Can x take some value such that x^5 is a perfect square? Say, x = 4. Then x^5 = 4^5 = 2^10 This is a perfect square. Hence for some value of x, x^5 could be a perfect square. Hence this is not our answer. How do we find a value for which x^5 will be a perfect square? Perfect squares have even powers. We have x^5 which is an odd power. To get an even power, we could select x such that it already has an even power - we selected x = 2^2. Similarly, x could be 1^2 or 3^2 or 4^2 or 5^2 or 3^4 etc Now think, can x^2 - 1 be a perfect square? The reasoning for all the options is given in the post above. A more intuitive approach is putting x = 1 as given by Bunuel. When x = 1, all options except option (D) results in a perfect square. So we know that all options CAN be perfect squares except (D). By elimination, answer must be (D). _________________ Karishma Veritas Prep | GMAT Instructor My Blog Save$100 on Veritas Prep GMAT Courses And Admissions Consulting
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Re: If x is a positive integer, which of the following CANNOT be [#permalink]  30 Mar 2014, 20:18
VeritasPrepKarishma wrote:
Mountain14 wrote:
Even though Bunuel and karishma have answered this question, I am not able to digest any fundamental of this ....

Not sure how to crack this one... as I got this question in my Veritas prep exam today...

Look, the question simply asks which option CANNOT be a perfect square. In the options, x is a positive integer.

Can x^5 be a perfect square? Can x take some value such that x^5 is a perfect square? Say, x = 4. Then x^5 = 4^5 = 2^10
This is a perfect square. Hence for some value of x, x^5 could be a perfect square. Hence this is not our answer.
How do we find a value for which x^5 will be a perfect square? Perfect squares have even powers. We have x^5 which is an odd power. To get an even power, we could select x such that it already has an even power - we selected x = 2^2. Similarly, x could be 1^2 or 3^2 or 4^2 or 5^2 or 3^4 etc

Now think, can x^2 - 1 be a perfect square?
The reasoning for all the options is given in the post above.

A more intuitive approach is putting x = 1 as given by Bunuel. When x = 1, all options except option (D) results in a perfect square. So we know that all options CAN be perfect squares except (D). By elimination, answer must be (D).

So the one point where I get tripped up is the X^2 - 1. 0 is assumed to be a perfect square?

How can I tell from the verbage of the question that what they are asking for is determining whether or not something is a perfect square or not?

Thanks for the help.
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Re: If x is a positive integer, which of the following CANNOT be [#permalink]  30 Mar 2014, 21:59
Expert's post
dbiersdo wrote:
VeritasPrepKarishma wrote:
Mountain14 wrote:
Even though Bunuel and karishma have answered this question, I am not able to digest any fundamental of this ....

Not sure how to crack this one... as I got this question in my Veritas prep exam today...

Look, the question simply asks which option CANNOT be a perfect square. In the options, x is a positive integer.

Can x^5 be a perfect square? Can x take some value such that x^5 is a perfect square? Say, x = 4. Then x^5 = 4^5 = 2^10
This is a perfect square. Hence for some value of x, x^5 could be a perfect square. Hence this is not our answer.
How do we find a value for which x^5 will be a perfect square? Perfect squares have even powers. We have x^5 which is an odd power. To get an even power, we could select x such that it already has an even power - we selected x = 2^2. Similarly, x could be 1^2 or 3^2 or 4^2 or 5^2 or 3^4 etc

Now think, can x^2 - 1 be a perfect square?
The reasoning for all the options is given in the post above.

A more intuitive approach is putting x = 1 as given by Bunuel. When x = 1, all options except option (D) results in a perfect square. So we know that all options CAN be perfect squares except (D). By elimination, answer must be (D).

So the one point where I get tripped up is the X^2 - 1. 0 is assumed to be a perfect square?

How can I tell from the verbage of the question that what they are asking for is determining whether or not something is a perfect square or not?

Thanks for the help.

Yes, both 0 and 1 are perfect squares.

"Can you express 'this' as n^2 where n is an integer?" asks us whether we can write 'this' as square of an integer.
Square of an integer is a perfect square. So the question becomes "Can you express 'this' as a perfect square?"

Slightly convoluted verbiage is common in GMAT.
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Re: If x is a positive integer, which of the following CANNOT be [#permalink]  20 Jul 2014, 00:05
Bunuel wrote:
emmak wrote:
If x is a positive integer, which of the following CANNOT be expressed as n^2, where n is an integer?

A. x^5
B. x^2 − 1
C. \sqrt{x^8}
D. x^2 + 1
E. \sqrt{x^5}

The question basically asks: if x is a positive integer, which of the following CANNOT be a perfect square.

Now, if x=1, then options A, B, C and E ARE perfect squares, therefore by POE the correct answer must be D.

Please refer to option [b] where value comes 0.
Would you please clarify whether 0 is a perfect square?
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Re: If x is a positive integer, which of the following CANNOT be [#permalink]  20 Jul 2014, 03:16
Expert's post
musunna wrote:
Bunuel wrote:
emmak wrote:
If x is a positive integer, which of the following CANNOT be expressed as n^2, where n is an integer?

A. x^5
B. x^2 − 1
C. \sqrt{x^8}
D. x^2 + 1
E. \sqrt{x^5}

The question basically asks: if x is a positive integer, which of the following CANNOT be a perfect square.

Now, if x=1, then options A, B, C and E ARE perfect squares, therefore by POE the correct answer must be D.

Please refer to option [b] where value comes 0.
Would you please clarify whether 0 is a perfect square?

Yes, 0 is a perfect square 0 = 0^2.
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Re: If x is a positive integer, which of the following CANNOT be   [#permalink] 20 Jul 2014, 03:16
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