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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] New post 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.
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Re: If x is a positive integer, which of the following CANNOT be [#permalink] New post 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.

Answer: D.
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Re: If x is a positive integer, which of the following CANNOT be [#permalink] New post 12 Feb 2014, 15:10
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Re: If x is a positive integer, which of the following CANNOT be [#permalink] New post 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] New post 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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Re: If x is a positive integer, which of the following CANNOT be [#permalink] New post 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.

Answer (D)
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Re: If x is a positive integer, which of the following CANNOT be [#permalink] New post 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... :(
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Re: If x is a positive integer, which of the following CANNOT be [#permalink] New post 30 Mar 2014, 18:01
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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).
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Re: If x is a positive integer, which of the following CANNOT be [#permalink] New post 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] New post 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] New post 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.

Answer: 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] New post 20 Jul 2014, 03:16
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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.

Answer: 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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NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

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RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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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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