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

Question

Competition Mode Question

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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freedom128 · Accepted Answer

x^2 is a positive integer, then x could be either positive or negative, as well as either integer or rational.

1) |x| is clearly an integer, but x can be either positive or negative.
NOT SUFFICIENT

2) |x|=x, so x is clearly positive. However, x can be a rational or an integer.
NOT SUFFICIENT

1)+2)
Surely, x is a positive integer.
SUFFICIENT

FINAL ANSWER IS (C)

CareerGeek · Answer

Given x^2 is a positive integer
--> x can be positive/negative integer or positive/negative irrational number
--> A possible set of values of x = ± 2 , ± \sqrt{2}

(1)  \sqrt{x^2}  is an integer.
--> Possible value of x = ±2 -->  Insufficient

(2)  \sqrt{x^2} = x 
--> 'x' has to be positive
--> Possible value of x =  2  or  \sqrt{2}  -->  Insufficient

Combining (1) & (2),
--> Possible value of x = 2 ONLY (Or any positive integer only) -->  Sufficient

Option C

exc4libur · Answer

x^2=positive.integer…|x|>0…x=

(1) √x^2 is an integer   insufic

\sqrt{x^2}=|x|=integer…x=

(2) √x^2=x   insufic

\sqrt{x^2}=x…\sqrt{x^2}=|x|=x…|x|=x…|x|>0…x>0…x=

(1&2)   sufic

x=integer>0

Ans (C)

lacktutor · Answer

If  x^{2}  is a positive integer, is x a positive integer?
(—> x cannot be equal to zero.)

(Statement1):  √x^{2}  is an integer.
—> x could be either positive or negative integer
If x=—3, then  √(—3)^{2}  =integer (No) 
If x= 3, then  √3^{2}  =integer (Yes) 
Insufficient

(Statement2):  √x^{2} = x  (—> x >0) 
If x= 3, then  √3^{2}  =3 (Yes) 
If x= √3, then  √(√3)^{2}  =√3 (No)
Insufficient

Taken together 1&2, 
  √x^{2} = x  and x is an integer.
Only if x is an positive integer, it satisfies the both statements 
Sufficient

The answer is C

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QuantMadeEasy · Answer

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

(1) √x^2 is an integer.
√x^2 = |x|
x can be positive or negative integer.
Insufficient

(2) √x^2=x
As √x^2 = |x|
Therefore, |x| = x; 
x is positive number, but may or may not be an integer.
Insufficient

(1)+(2), x is an integer which is positive
Sufficient

C is correct

debjit1990 · Answer

Ans: C

x^2 is a positive integer-->> 5,9 etc
(1) √x2 is an integer...so x^2 cannot be 5
it can be 9..so x= +3 or -3. not sufficient

2)√x2=x, so x can be √5 or 3. not sufficient

combined-->> x can only be integer and positive

firas92 · Answer

If  x^2  is positive,  x  is not equal to 0

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

\sqrt{x^2} = |x| =integer

If  |x|  is an integer and  x  is not 0,  |x|  must be a positive integer

So  x  is a positive integer if  |x|=x

x  is a negative integer if  |x|=-x

1 is not sufficient

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

Here,  |x|=x  so  x  must be positive but we don't know if it is an integer

2 is insufficient

(1)+(2)

|x|  is an integer and  |x|=x  so  x  must be a positive integer

Sufficient

Answer is  (C)

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Staphyk · Answer

Damn it! subtle stuff,was all about integers and non-Integer. Lolx why I love this 700-level Questions

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unraveled · Answer

Exact emotions here as well

. Somewhere while solving had a hunch that i was going wrong.

Bharti_Pandit · Answer

If x^2 is an integer shouldn't always be an integer?

Bunuel · Answer

For x^2 to be an integer, x could either be an integer or an irrational number that is the square root of an integer, such as  \sqrt{2} ,  \sqrt{3} , etc. Therefore, x^2 being an integer does not necessarily mean that x is an integer.

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

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GMAT Data Sufficiency (DS) Questions

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

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