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# If x2 is a positive integer, is x a positive integer?

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Senior Manager
Joined: 08 Jun 2013
Posts: 469
Location: India
GMAT 1: 200 Q1 V1
GPA: 3.82
WE: Engineering (Other)
If x2 is a positive integer, is x a positive integer?  [#permalink]

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15 Sep 2018, 18:38
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Difficulty:

65% (hard)

Question Stats:

18% (00:37) correct 82% (01:06) wrong based on 55 sessions

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If $$x^2$$ is a positive integer, is $$x$$ a positive integer?

(1) $$\sqrt{x^2}$$ is an integer.

(2) $$\sqrt{x^2} = x$$

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Joined: 02 Aug 2009
Posts: 7035
Re: If x2 is a positive integer, is x a positive integer?  [#permalink]

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15 Sep 2018, 20:30
3
If $$x^2$$ is a positive integer, is $$x$$ a positive integer?
$$x/\neq{0}$$

(1) $$\sqrt{x^2}$$is an integer.
If x is -3, $$\sqrt{(-3)^2}$$=3
Same if x is 3..
Insufficient

(2) $$\sqrt{x^2}$$ = $$x$$
x>0
Say $$x^2=3........\sqrt{x^2}=\sqrt{3}.......x=√3$$...x is positive but not integer
But x=3....xis positive and integer
Insufficient

Combined..
x>0...
And if x>0, then √(x^2) can be integer only when x is integer..
So x is positive and also integer
Sufficient

C
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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Re: If x2 is a positive integer, is x a positive integer?  [#permalink]

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16 Sep 2018, 09:13
(1) $$\sqrt{x^2}$$ is an integer. --> x is an integer. But we don't know whether x is positive or negative. --> Not sufficient.
Number plugging:
If $$\sqrt{3^2}= 3$$ --> $$x=3$$ --> Yes.
If $$\sqrt{(-3)^2}= 3$$ --> $$x=-3$$ --> No.

(2) $$\sqrt{x^2} = x$$ --> x is positive. But we don't know whether x is integer. --> Not sufficient.
Number plugging:
If $$\sqrt{3^2}= 3$$ --> $$x=3$$ --> Yes.
If $$\sqrt[]{(\sqrt[]{3})^{2}} = \sqrt[]{3}$$ --> $$x=\sqrt[]{3}$$ --> No.

(1) + (2): x is an integer AND x is positive.--> Yes --> Sufficient.

Harshgmat wrote:
If $$x^2$$ is a positive integer, is $$x$$ a positive integer?

(1) $$\sqrt{x^2}$$ is an integer.

(2) $$\sqrt{x^2} = x$$

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Re: If x2 is a positive integer, is x a positive integer? &nbs [#permalink] 16 Sep 2018, 09:13
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