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

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

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15 Sep 2018, 19:38
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Difficulty:

85% (hard)

Question Stats:

21% (01:00) correct 79% (01:16) wrong based on 111 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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Re: If x2 is a positive integer, is x a positive integer?  [#permalink]

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15 Sep 2018, 21:30
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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Re: If x2 is a positive integer, is x a positive integer?  [#permalink]

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16 Sep 2018, 10:13
(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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If x2 is a positive integer, is x a positive integer?  [#permalink]

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10 Dec 2018, 02:56
In statement (2) ,

why can't X be negative?
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Re: If x2 is a positive integer, is x a positive integer?  [#permalink]

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10 Dec 2018, 03:04
LHS is square root of a real number => LHS is not negative.

Hope it's clear.

Pritam_Pf wrote:
In statement (2) ,

why can't X be negative?

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

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10 Dec 2018, 03:42
x^2 has to be +ve as it's under square root. This does not mean that x has to be +ve.x canbe -ve.

So correct answer should be E.

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Re: If x2 is a positive integer, is x a positive integer?   [#permalink] 10 Dec 2018, 03:42
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