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

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Intern
Joined: 28 May 2013
Posts: 2
Re: If x is a positive integer, is x^1/2 an integer [#permalink]

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28 May 2013, 00:26
1
Hi all,
I have a question solution from the official book that appears incorrect to me.
I've added a scan as an attachment.

The solution claims that A is sufficient. But you don't always get an integer, for example:
If you take \sqrt{4x}=5 vs \sqrt{4x}=6.

what do you think?
Attachments

gmat.JPG [ 31.55 KiB | Viewed 17710 times ]

Math Expert
Joined: 02 Sep 2009
Posts: 47037
Re: If x is a positive integer, is x^1/2 an integer [#permalink]

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28 May 2013, 00:41
shimonam wrote:
Hi all,
I have a question solution from the official book that appears incorrect to me.
I've added a scan as an attachment.

The solution claims that A is sufficient. But you don't always get an integer, for example:
If you take \sqrt{4x}=5 vs \sqrt{4x}=6.

what do you think?

All OG13 questions with solutions are here: the-official-guide-quantitative-question-directory-143450.html

Hope it helps.
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Joined: 02 Sep 2009
Posts: 47037
Re: If x is a positive integer, is root(x) an integer? [#permalink]

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05 Jun 2013, 03:55
Bumping for review and further discussion*. Get a kudos point for an alternative solution!

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Re: If x is a positive integer, is root(x) an integer? [#permalink]

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05 Jun 2013, 07:58
[quote="alexBLR"]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) since $$\sqrt{4x}$$ is an integer, it is the same as 2*$$\sqrt{x}$$ that means $$\sqrt{x}$$ also must be ans integer - sufficient

(2) $$\sqrt{3x}$$ is not an integer, in this case $$\sqrt{x}$$ could be or couldn't be an integer.
Case 1) x=4, where $$\sqrt{x}$$ is an integer, then $$\sqrt{3*4}$$=2$$\sqrt{3}$$ still not an integer
Case 2) x=5, where $$\sqrt{x}$$ is NOT an integer, then $$\sqrt{3*5}$$=$$\sqrt{15}$$ still not an integer.
This statement is not sufficient. Answer is A
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Re: If x is a positive integer, is x^1/2 an integer [#permalink]

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29 Jun 2013, 06:03
The square root of any positive integer is either an integer or an irrational number. So, $$\sqrt{x}=\sqrt{integer}$$ cannot be a fraction, for example it cannot equal to 1/2, 3/7, 19/2, ... It MUST be an integer (0, 1, 2, 3, ...) or irrational number (for example $$\sqrt{2}$$, $$\sqrt{3}$$, $$\sqrt{17}$$, ...).

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Re: If x is a positive integer, is root(x) an integer? [#permalink]

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27 Dec 2013, 06:37
alexBLR wrote:
If x is a positive integer, is $$\sqrt{x}$$ an integer?

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

This is the question from GMAT Quant Review. My logic to solve this question:

\sqrt{4x}=2*\sqrt{x}, so \sqrt{x} can either be integer or not an integer (for example \sqrt{x}=2.5) and the 2*\sqrt{x} is still an integer. So Statement 1 is insufficient.

\sqrt{3x}= \sqrt{3}*\sqrt{x}. As \sqrt{3} is not an integer, the \sqrt{x} can be either integer or non integer and the \sqrt{3}*\sqrt{x} will still be not integer. So Statement 2 is insufficient.

S1 and S2 together is still insufficient as \sqrt{x}=2 and \sqrt{x}=2.5 both satisfy statements requirement.

So I choose E as an answer.

Is there a flaw in my reasoning? OG Quant review answer to this question is different from E.

From First statement we get 2 sqrt (x) is an integer
Therefore sqrt (x) must be an integer

Just to elaborate a bit further:

The thing is that sqrt (x) can't be an integer/2, cause the square root of any positive integer (as stated in the givens) is always >=1. Therefore, sqrt (x) will have to be an integer.

Suff

From second setatement we get the sqrt (3x) is not an integer
Therefore, 3x is not a perfect square
or x is not perfect square that is multiple of 3
But it could very well be a perfect square
Matter is we still don't know

Just to clarify, sqrt (x) could or could not be an integer. If sqrt root (x) is an integer then sure 1.7 (integer) not an integer. But, sqrt (x) could also be say x=2, therefore sqrt (2)=1.4. Then 1.4*1.7 is clearly not an integer either. Answer A stands

Insuff

Hence A is our best choice here

Hope it helps
Let the kudos rain begin

Cheers!
J
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Joined: 23 Aug 2014
Posts: 37
GMAT Date: 11-29-2014
Re: If x is a positive integer, is x^1/2 an integer [#permalink]

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12 Nov 2014, 01:36
Hello
Easiest way i can think of:
S1-$$\sqrt{4x}$$ is an integer =$$\sqrt{4}*\sqrt{x}$$ has to be integer
We know that $$\sqrt{4}$$ is 2 - int.
so$$\sqrt{x}$$ has to be int ( because Non Integer*integers is ALWAYS a Non integer, which is not the case here)
Sufficient (12TEN) or (ABCDE)

S2- $$\sqrt{3x}$$ is NOT an integer = $$\sqrt{3}*\sqrt{x}$$ is not an integer
We know that $$\sqrt{3}$$ is not an integer
But Non-Integer * Integer OR Non-integer = Non- integer

thoughts?
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Re: If x is a positive integer, is x^1/2 an integer [#permalink]

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05 Dec 2014, 00:00
deeuk wrote:
Hello
Easiest way i can think of:
S1-$$\sqrt{4x}$$ is an integer =$$\sqrt{4}*\sqrt{x}$$ has to be integer
We know that $$\sqrt{4}$$ is 2 - int.
so$$\sqrt{x}$$ has to be int ( because Non Integer*integers is ALWAYS a Non integer, which is not the case here)
Sufficient (12TEN) or (ABCDE)

S2- $$\sqrt{3x}$$ is NOT an integer = $$\sqrt{3}*\sqrt{x}$$ is not an integer
We know that $$\sqrt{3}$$ is not an integer
But Non-Integer * Integer OR Non-integer = Non- integer

thoughts?

Bunuel could you please tell me if this approach is correct/incorrect?
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Posts: 47037
Re: If x is a positive integer, is x^1/2 an integer [#permalink]

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05 Dec 2014, 04:28
Motivatedtowin wrote:
deeuk wrote:
Hello
Easiest way i can think of:
S1-$$\sqrt{4x}$$ is an integer =$$\sqrt{4}*\sqrt{x}$$ has to be integer
We know that $$\sqrt{4}$$ is 2 - int.
so$$\sqrt{x}$$ has to be int ( because Non Integer*integers is ALWAYS a Non integer, which is not the case here)
Sufficient (12TEN) or (ABCDE)

S2- $$\sqrt{3x}$$ is NOT an integer = $$\sqrt{3}*\sqrt{x}$$ is not an integer
We know that $$\sqrt{3}$$ is not an integer
But Non-Integer * Integer OR Non-integer = Non- integer

thoughts?

Bunuel could you please tell me if this approach is correct/incorrect?

No. That's not correct.

(Non-integer)*(integer) could yield an integer. For example, 1/2*2 = 1.

(Non-integer)*(non-integer) could also yield an integer. For example, $$\sqrt{2}*\sqrt{2}=2$$.
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Re: If x is a positive integer, is x^1/2 an integer [#permalink]

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10 Dec 2014, 13:21
Thank you for the correction
But now i am wondering since $$\sqrt{4}$$ is 2, doesn't $$\sqrt{x}$$ have to be an integer as well. I mean doesnt x have to be a perfect square?
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Re: If x is a positive integer, is x^1/2 an integer [#permalink]

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11 Dec 2014, 03:35
my approach

I try to be careful in such types of questions because of a trap where a fraction multiplied by an integer can lead to an integer.

Statement 1
sqrt(4x)= +-2 * sqrt(x) (-2 not for gmat)
so sqrt(4x)= 2*sqrt(x)
here sqrt(x) has to be an integer or an irrational no. no matter what form x is in (complex nos out of scope so ignore)
as per the statement 1 it will be an integer. so sufficient.

now lets check the trap
if sqrt(x) is of the form integer/2 then still statement 1 will hold true but since x is an integer then at best x will of the form 4*perfect square. so again this statement will also give us a positive answer.

Statement 2
this basically gives us two opposite answers
we need to compare x when it is a
1. perfect square (e.g. 25, sqrt(x)=5 and statement 2 is true)
2. not a perfect square (e.g 10 sqrt(10)=3.16 and statement 2 is still true)
can't deduce anything
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J

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Re: If x is a positive integer, is x^1/2 an integer [#permalink]

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27 Sep 2015, 08:36
Bunuel wrote:
ezinis wrote:
If x is a positive integer, is \sqrt{x} an integer?
(1) $$\sqrt{4x}$$ is an integer4.
(2) $$\sqrt{3x}$$ is an integer.

I am not satisfied with the official explanation. Please give yours, thanks.

I think (2) should be $$\sqrt{3x}$$ is NOT an integer.

If $$x=integer$$, is $$\sqrt{x}=integer$$?

(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.

i think it does not matter whether statement 2 is integer or not, as in both the cases we are getting different solution. and answer will be one in both the cases.
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Re: If x is a positive integer, is root(x) an integer? [#permalink]

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24 Dec 2015, 04:30
sqrt (4x) is an integer implies that 2 * sqrt(x) is also an integer. That implies x is an integer. Statement 1 is sufficient

sqrt (3x) is not an integer. For x = 2 it is possible. For a number such as 4 also it is. But sqrt () is not an integer while sqrt (4) is. Insufficient

Hence A
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Re: If x is a positive integer, is x^1/2 an integer [#permalink]

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12 Feb 2017, 14:41
I am new to gmatclub so I apologize if I am using this forum incorrectly... I instinctively want to plug in numbers: The way i solved was:
Sqrt(4*1)= 2= integer but sqrt(1) is not an integer
Sqrt(4*4)= 4= integer and sqrt(4)=2=integer

so I would say statement 1 is not sufficient. Clearly my logic is flawed since statement 1 is sufficient, but can someone tell me why making up numbers for X does not work for this problem?
Math Expert
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Posts: 47037
Re: If x is a positive integer, is x^1/2 an integer [#permalink]

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12 Feb 2017, 14:44
1
gzimmer wrote:
I am new to gmatclub so I apologize if I am using this forum incorrectly... I instinctively want to plug in numbers: The way i solved was:
Sqrt(4*1)= 2= integer but sqrt(1) is not an integer
Sqrt(4*4)= 4= integer and sqrt(4)=2=integer

so I would say statement 1 is not sufficient. Clearly my logic is flawed since statement 1 is sufficient, but can someone tell me why making up numbers for X does not work for this problem?

Highlighted part is not correct. $$\sqrt 1 = 1 = integer$$
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Re: If x is a positive integer, is x^1/2 an integer [#permalink]

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12 Feb 2017, 15:00
wow, that was a dumb mistake on my part. thanks Bunuel !!
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Re: If x is a positive integer, is x^1/2 an integer [#permalink]

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25 Nov 2017, 00:05
OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/if-x-is-a-po ... 65976.html
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Re: If x is a positive integer, is x^1/2 an integer   [#permalink] 25 Nov 2017, 00:05

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