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# GMAT Diagnostic Test Question 3

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GMAT Diagnostic Test Question 3 [#permalink]

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06 Jun 2009, 19:33
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GMAT Diagnostic Test Question 3
Field: Arithmetic, Roots
Difficulty: 700

If $$x$$ is a positive number and $$\sqrt{x} * x - x = a$$, which of the following must be true?

I. $$a$$ is Even
II. $$a$$ is Positive
III. $$a$$ is an Integer

A. I only
B. II only
C. III only
D. I and II
E. None of the above
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Last edited by Bunuel on 07 Oct 2013, 00:04, edited 3 times in total.
Edited the stem and difficulty.
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Re: GMAT Diagnostic Test Question 3 [#permalink]

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06 Jun 2009, 19:35
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Explanation:

Note that we are asked "which of the following MUST be true, not COULD be true. For such kind of questions if you can prove that a statement is NOT true for one particular set of numbers, it will mean that this statement is not always true and hence not a correct answer.

If $$x=\frac{1}{4}$$ then $$a=\sqrt{x} * x - x=\frac{1}{2}*\frac{1}{4}-\frac{1}{4}=-\frac{1}{8}$$. Now, $$-\frac{1}{8}$$ is not an integer at all (hence not even) and also not positive, so none of the options MUST be true.

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Last edited by bb on 28 Sep 2013, 11:34, edited 1 time in total.
Updated
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Re: GMAT Diagnostic Test Question 3 [#permalink]

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12 Aug 2009, 02:07
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Good one, +1.
trangpham wrote:
I think the correct answer is "E" because if x=1, a=0 and 0 is not a positive.
If x=2, $$\sqrt{2} - 1$$ is not a integer or even.

ihero06, you seem to forget about $$x=0$$. If $$x=0$$, $$a$$ is not positive. The question asks "which of the following must be true", meaning for all possible $$x$$. You're right that GMAT doesn't cover negative numbers under the radical sign. We have to remember that 0 is neither positive nor negative.

I've updated the OE a bit. Hopefully it's better now.
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Re: GMAT Diagnostic Test Question 3 [#permalink]

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19 Dec 2009, 15:45
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I found the key to this question is "must be true" clause.

Meaning if we can prove that a could be a non-Integer (i.e. x = 2) or 0 then we rule out all answers but E.

Good question!
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Re: GMAT Diagnostic Test Question 3 [#permalink]

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04 Aug 2009, 14:01
I think the answer could be B (II -- a is positive) if we consider sqrt of 2 and 3 as irrational (or sqrt of any other prime number).

or D (I and II -- a is positive and even). Anyway, I feel that the specification of x is not fully given.

The Q is
\sqrt{x}*x - x = a

and x is an integer.

If x is -ve, then we have an equation contains sqrt of negative integer- which is not covered in GMAT. So let's rule out -ve integer. If x is -ve, then the answer is "E- none of the above"

If x is positive then:
If x is even (E), then
SQRT(E)*E is even and E-E is also even. and so a is even and it is positive.

x= 2 then 2*sqrt(2)-2 = not integer, but if we consider sqrt (2) as an irrational number then a has to be positive.
x= 16 then 16*sqrt(16)-16 = 16*4-16 = 48 - +ve and even.

If x is odd (O), then
SQRT(O)*O is odd and O-O is even. So a is even and again it is positive.
x=3, then 3*sqrt(3)-3 = not integer, but a has to be positive again {sqrt (3) is irrational}.
x=9, then 9*sqrt(9)-9 = 18 --> +ve and even.

So if one considers sqrt of 2 and 3 are irrational numbers then we can have choice B (a is positive).
Otherwise, among the choices the answer is D (I & II - i.e. a is even and positive).

Let me know whether I am correct or not.
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Re: GMAT Diagnostic Test Question 3 [#permalink]

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10 Aug 2009, 19:04
I think the correct answer is "E" because if x=1, a=0 and 0 is not a positive.
If x=2, $$\sqrt{2} - 1$$ is not a integer or even.
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Re: GMAT Diagnostic Test Question 3 [#permalink]

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16 Oct 2009, 05:48
Should we rule out the possibilty of X != -1 if it is not explicitly mentioned here?
What is the final answer..I will go with E..
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Re: GMAT Diagnostic Test Question 3 [#permalink]

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16 Oct 2009, 10:00
The answer is E. It's stated in the second post.
KocharRohit wrote:
Should we rule out the possibilty of X != -1 if it is not explicitly mentioned here?
What is the final answer..I will go with E..

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Re: GMAT Diagnostic Test Question 3 [#permalink]

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18 Oct 2009, 07:33
As stated by the founder, the key of this question is to show that a is not an integer, if so, a non integer number can not be even nor odd and therefore one is left to prove whether a is positive or negative. Since 0 is and integer and mathematically speaking it has not sign convention i.e. zero has no distance, then given that in this case a = 0, and one can not tell what the sign of a is, then the right answer must be E.
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Re: GMAT Diagnostic Test Question 3 [#permalink]

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11 Jan 2010, 22:00
bb wrote:
Explanation:
 Rating:

If we prove that $$a$$ is not an integer for any $$x$$, we'll be able to rule out A, C, and D. $$\sqrt{x} * x - x = a$$ can be rewritten as $$x(\sqrt{x}-1)=a$$. If $$x=2$$ the expression turns into $$2(\sqrt{2}-1)=a$$. It's clear that $$a$$ is not an integer as $$(\sqrt{2}-1)$$ is not an integer. Now we need to prove that $$a$$ is not necessarily positive. If $$x=0$$, the equation turns $$0=a$$. 0 is neither positive, nor negative. We're left with E as the answer.

Thanks for the explanation. Got correct answer but did more work than necessary. 1) Didn't rephrase; 2) plugged in 0, 1, 4, and 3. Had I just plugged in 2 and 0 (with the intention of ruling out answer choices), would have gotten to E much faster. Didn't spend a lot of time on it. Nonetheless, my goal should be to get the correct answer as quickly as possible.
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Re: GMAT Diagnostic Test Question 3 [#permalink]

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16 Feb 2010, 07:38
LUGO wrote:
As stated by the founder, the key of this question is to show that a is not an integer, if so, a non integer number can not be even nor odd and therefore one is left to prove whether a is positive or negative. Since 0 is and integer and mathematically speaking it has not sign convention i.e. zero has no distance, then given that in this case a = 0, and one can not tell what the sign of a is, then the right answer must be E.

i always thought zero was positive...since it was 1 less than "1"
anyhow...thanks for this note.
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Re: GMAT Diagnostic Test Question 3 [#permalink]

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02 Apr 2010, 04:42
1st I factorized: x(\sqrt{x} - 1) = a
the stem did not specified whether x is +ve /-ve; odd or even integer.
\sqrt{9} - 1 => even; but \sqrt{1/9) - 1 = 1/3 - 1 = -2/3
=> the result could be either even, rational, or -ve. So, I, II & III out.
Hence, E.
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Re: GMAT Diagnostic Test Question 3 [#permalink]

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23 Apr 2010, 09:25
What if x=25 or x=36 ?
than : 25*(5-1)=100 - even
36*(6-1)=180 - even

based on these examples I have been lured away.
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Re: GMAT Diagnostic Test Question 3 [#permalink]

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09 Feb 2011, 05:03
I think the best solution is :

if u notice in the question it is written that x is an integer i.e it can have any value from minus infinity to plus infinity.
now if we take x as a negative number (for ex : -5)then the solution turns out be
a=-5*sqrt(-5)+(-5)

we allknow that sqrt of -ve number is an imaginary value which can neither be positive ,nor even nor an integer.

Artabandhu(Its my first post feel free to post comments)
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Re: GMAT Diagnostic Test Question 3 [#permalink]

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24 May 2011, 14:39
Is it also possible to solve this one without factoring? \sqrt{x} * x - x = a and say for every x you put in you'd have to multiply it with zero thus making the outcome none of the one mentioned in the answers? :S
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Re: GMAT Diagnostic Test Question 3 [#permalink]

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28 Sep 2011, 11:27
if x = 4 we get sqrt of 4 x 4 - 4 = 2 x 4 - 4 = 8 - 4 = 4. Which is positive, even and an integer. Am i missing something? I saw you guys trying it with 2 and 3 and 0, but when you try it with an integer that has a sqrt root then you dont get this answer... can SOME PLEASE ASSIST?
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Re: GMAT Diagnostic Test Question 3 [#permalink]

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28 Sep 2011, 11:33
Expert's post
Djdieg007 wrote:
if x = 4 we get sqrt of 4 x 4 - 4 = 2 x 4 - 4 = 8 - 4 = 4. Which is positive, even and an integer. Am i missing something? I saw you guys trying it with 2 and 3 and 0, but when you try it with an integer that has a sqrt root then you dont get this answer... can SOME PLEASE ASSIST?

The answer has to work in 100% of the situation, so even though 4 works, the equation does not work with others, and therefore, the answer is E. You have to try multiple numbers and if even one of them works, it is out.
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Re: GMAT Diagnostic Test Question 3 [#permalink]

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29 Sep 2011, 21:45
Djdieg007 wrote:
if x = 4 we get sqrt of 4 x 4 - 4 = 2 x 4 - 4 = 8 - 4 = 4. Which is positive, even and an integer. Am i missing something? I saw you guys trying it with 2 and 3 and 0, but when you try it with an integer that has a sqrt root then you dont get this answer... can SOME PLEASE ASSIST?

HERE, you have considered only one case.

answer should fulfill all the case.
Consider X= 2 , 8 ,10, all are even but for these values, a is neither odd nor Even. a is not even an integer.
same is applicable if you consider X = 2,3,5,6,7,8,10,11,12,13,14,15....
you can generalize it by saying that any value of X, which is not perfect square irrespective of odd or even , will yield a as decimal form (not Integer).

So, the Ans is E.

Hope it ll help.
you can post here still you have any doubt.
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Re: GMAT Diagnostic Test Question 3 [#permalink]

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04 Oct 2011, 05:48
bb wrote:
GMAT Diagnostic Test Question 3
Field: Arithmetic, Roots
Difficulty: 700-750
 Rating:

If $$x$$ is an integer and $$\sqrt{x} * x - x = a$$ , which of the following must be true?
I. $$a$$ is Even
II. $$a$$ is Positive
III. $$a$$ is an Integer

A. I only
B. II only
C. III only
D. I and II
E. None of the above

$$x (\sqrt{x} -1) = a$$

I. a is even when X= 9, but a is fraction when X=5 (say, $$5 (\sqrt{5} -1)$$. So no
II. a is positive when X=9, but a is "0" when X=1. so no
III. a is not an integer when X=5. so no

E is my pick.
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Re: GMAT Diagnostic Test Question 3 [#permalink]

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28 Oct 2011, 22:38
Take x=1 => 1\sqrt{1}-1=0

Since 0 is neither +ve nor -ve so option b is out.

now take x=2 => 2\sqrt{2}-2 => neither an integer nor an even.

So ans=E
Re: GMAT Diagnostic Test Question 3   [#permalink] 28 Oct 2011, 22:38

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# GMAT Diagnostic Test Question 3

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