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If x is a positive integer, is x^1/2 an integer

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If x is a positive integer, is x^1/2 an integer [#permalink] New post 13 Jan 2014, 00:51
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The Official Guide For GMAT® Quantitative Review, 2ND Edition

If x is a positive integer, is \(\sqrt{x}\) an integer?

(1) \(\sqrt{4x}\) is an integer.
(2) \(\sqrt{3x}\) is not an integer.

Data Sufficiency
Question: 31
Category: Algebra Radicals
Page: 155
Difficulty: 650


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Re: If x is a positive integer, is x^1/2 an integer [#permalink] New post 13 Jan 2014, 00:52
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SOLUTION:

If x is a positive integer, is \(\sqrt{x}\) an integer?

As given that \(x\) is a positive integer then \(\sqrt{x}\) is either an integer itself or an irrational number.

(1) \(\sqrt{4x}\) is an integer --> \(2\sqrt{x}=integer\) --> \(2\sqrt{x}\) to be an integer \(\sqrt{x}\) must be an integer or integer/2, but as \(x\) is an integer, then \(\sqrt{x}\) can not be integer/2, hence \(\sqrt{x}\) is an integer. Sufficient.

(2)\(\sqrt{3x}\) is not an integer --> if \(x=9\), condition \(\sqrt{3x}=\sqrt{27}\) is not an integer satisfied and \(\sqrt{x}=3\) IS an integer, BUT if \(x=2\), condition \(\sqrt{3x}=\sqrt{6}\) is not an integer satisfied and \(\sqrt{x}=\sqrt{2}\) IS NOT an integer. Two different answers. Not sufficient.

Answer: A.

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Re: If x is a positive integer, is x^1/2 an integer [#permalink] New post 14 Jan 2014, 06:06
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IMO A.

If x is a positive integer, is \sqrt{x} an integer?

(1) \sqrt{4x} is an integer.
(2) \sqrt{3x} is not an integer.

1) \sqrt{4x} = 2\sqrt{x} is also integer = hence sufficient.

2) if x = 27 then Yes,
if x = 4 then no 2. hence not sufficient.
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Re: If x is a positive integer, is x^1/2 an integer [#permalink] New post 14 Jan 2014, 10:49
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Bunuel wrote:
The Official Guide For GMAT® Quantitative Review, 2ND Edition

If x is a positive integer, is \(\sqrt{x}\) an integer?

(1) \(\sqrt{4x}\) is an integer.
(2) \(\sqrt{3x}\) is not an integer.



Statement 1) root(4x) is an integer. root(4x) = either +2root(x) or -2root(x) . As in either case, the result is an integer, then root(x) has to be an integer as (+2 or -2) is integer. Hence Sufficient.
Statement 2) root(3x) is an integer. root(3) is not an integer.
There can be following two cases:
1) root(x) is an integer => root(3) * root(x) is not an integer as root(3) is not an integer.
2) root(x) is not an integer => root(3) * root(x) can never be an integer except in once case where x= 3.
Either case it is not sufficient.

Hence Option A)
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Re: If x is a positive integer, is x^1/2 an integer [#permalink] New post 17 Jan 2014, 02:49
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SOLUTION:

If x is a positive integer, is \(\sqrt{x}\) an integer?

As given that \(x\) is a positive integer then \(\sqrt{x}\) is either an integer itself or an irrational number.

(1) \(\sqrt{4x}\) is an integer --> \(2\sqrt{x}=integer\) --> \(2\sqrt{x}\) to be an integer \(\sqrt{x}\) must be an integer or integer/2, but as \(x\) is an integer, then \(\sqrt{x}\) can not be integer/2, hence \(\sqrt{x}\) is an integer. Sufficient.

(2)\(\sqrt{3x}\) is not an integer --> if \(x=9\), condition \(\sqrt{3x}=\sqrt{27}\) is not an integer satisfied and \(\sqrt{x}=3\) IS an integer, BUT if \(x=2\), condition \(\sqrt{3x}=\sqrt{6}\) is not an integer satisfied and \(\sqrt{x}=\sqrt{2}\) IS NOT an integer. Two different answers. Not sufficient.

Answer: A.

Similar questions:
if-x-is-a-positive-integer-is-sqrt-x-an-integer-88994.html
value-of-x-107195.html
number-prop-ds-106886.html
number-system-106606.html
odd-vs-even-trick-question-106562.html
quant-review-2nd-edition-ds-104421.html
algebra-ds-101464.html
quant-review-2nd-edition-ds-104421.html
q-31-og-12-ds-101918.html
airthmetic-ds-108287.html
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If x is a positive integer, is x^1/2 an integer [#permalink] New post 18 Jan 2014, 03:22
If x is a positive integer, is \(\sqrt{x}\) an integer?

(1)\(\sqrt{4x}\) is an integer.
(2)\(\sqrt{3x}\) is not an integer.



Statement 1

\(\sqrt{4x} = k\), \(k\) is an integer
Or,\(\sqrt{x}= k/2\)

If \(k/2\) is not an integer, then \((k/2)^2\) or \(k^2/4\) is also not an integer and this implies that x is not an integer as \(x = k^2/4\).
But, it is given that \(x\) is an integer.

Therefore, \(k^2/4\) is an integer => \(k/2\) is an integer =>\(\sqrt{x}\) is an integer...........Sufficient....(B)(C)(E)


Statement 2

\(\sqrt{3x}=k\), \(k\) is not an integer
Or, \(\sqrt{x} = k/\sqrt{3}\)

Thus, \(\sqrt{x}\) = integer, when \(k\) is a multiple of \(\sqrt{3}\)
and \(\sqrt{x}\) is not an integer when \(k\) is not a multiple of \(\sqrt{3}\)

As the value of \(\sqrt{x}\) can not be uniquely determined, statement (2) is not sufficient.................(D)

Answer: (A)

Last edited by arunspanda on 31 Jul 2014, 15:18, edited 1 time in total.
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DS- Looking for a different solution [#permalink] New post 31 Jul 2014, 13:53
The following is a Question from the GMAT OG Quant review book 2nd Ed, DS Sample question no. 31.

The Answer explanation provided went over my head.

If x is a positive integer, is \sqrt{}X an integer?
(1) \sqrt{}4X is an integer.
(2) \sqrt{}3X is not an integer.

All you good people out there.. please provide me with a different - layman explanation.
Thank you in advance
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Re: If x is a positive integer, is x^1/2 an integer [#permalink] New post 31 Jul 2014, 13:58
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prada17 wrote:
The following is a Question from the GMAT OG Quant review book 2nd Ed, DS Sample question no. 31.

The Answer explanation provided went over my head.

If x is a positive integer, is \sqrt{}X an integer?
(1) \sqrt{}4X is an integer.
(2) \sqrt{}3X is not an integer.

All you good people out there.. please provide me with a different - layman explanation.
Thank you in advance


Merging topics. Please refer to the solution above.

P.S. Please read carefully and follow: rules-for-posting-please-read-this-before-posting-133935.html Pay attention to rules 1, 2 and 3. Thank you.


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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, is x^1/2 an integer   [#permalink] 31 Jul 2014, 13:58
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