If x is an integer such that x 0, is it true that x is an integer?

Question

If x is an integer such that x > 0, is it true that  \sqrt{x}  is an integer?

(1)  9* \sqrt{16x}  is an integer.

(2)  \sqrt{6x}  is not an integer.

GMATinsight · Answer

Given:  x > 0 and x is a positive integer

Question : Is it true that  \sqrt{x}  is an integer?
OR
  Question Redefined  : Is x a Perfect Square?

Statement 1 :  9* \sqrt{16x}  is an integer.

9* \sqrt{16x}  =  9*4* \sqrt{x}  = Integer
i.e.  \sqrt{x}  must be an Integer for an Integer result
  SUFFICIENT

Statement 2 :  \sqrt{6x}  is not an integer. 
Case-1:  \sqrt{x} = \sqrt{2}  Satisfies the statement and x is NOT a Perfect Square
Case-2:  \sqrt{x} = 2  Satisfies the statement and x=4 i.e. x is a Perfect Square
 NOT SUFFICIENT

Answer: Option A

ENGRTOMBA2018 · Answer

Given x>0 and is an integer. Is  \sqrt{x}  = integer? This can only happen if x is a perfect sqaure (=1,4,9, etc)

1. Per statement 1,  9* \sqrt{16x}  = integer ---->>  9* 4\sqrt{x} = integer--->  \sqrt{x}  = integer and is hence sufficient.

2. Per statement 2,  \sqrt{6x}  is NOT an integer. ----> Let x = 1 ,  \sqrt{6} = not an integer and if x= 3,  18 = not an integer , Thus we see  \sqrt{x}  may or may not be an integer. This statement is not sufficient to answer the question.

A is the correct answer.

ENGRTOMBA2018 · Answer

GMATinsight , how can  \sqrt{x}  =1/2 when it is given that x = integer?

chetan2u · Answer

(1)  9* \sqrt{16x}  is an integer.

9* 4\sqrt{x}  is an integer...
therefore  \sqrt{x}  will be an integer..suff

2)  \sqrt{6x}  is not an integer...
as  \sqrt{6}  is not an integer, so   \sqrt{x}   can be an integer or non integer and still satisfy the condition... insuff

ans A

Sindbad · Answer

from(1)
9∗√(16x) is an integer. => 9*4√x is an Integer=> 36√x is an integer=> this is only possible when x is a perfect square and hence √x belongs to Integer  Sufficient

from(2)
√(6x) is not an integer=> it can be x is a perfect square hence √(6x) is not or it can be x is not a perfect square .There can be multiple scenarios and multiple answers.  hence Not Sufficient

Answer: A

Bunuel · Answer

800score Official Solution:

Let’s evaluate each of the statements independently first. The question assumes that x is a positive integer. In order for (1) to be true, it should be obvious to the reader that x must be a perfect square. In other words we can break down (1), using the well-known properties of roots. Now, if 4√x is an integer, it must be true that √x is in fact an integer; there is no way that the product of an integer and an irrational number can give an integer. So, statement (1) is sufficient.

Now, let’s look at Statement (2). Sometimes it will answer the question, other times it won’t. For example, if x = 9 then √54 is not an integer. Yet, √9 = 3 is an integer. On the other hand, if x = 2 then √12 is not an integer and neither is√2. So, Statement (2) is insufficient, and the only reasonable  answer choice is (A) .

If x is an integer such that x > 0, is it true that x is an integer?

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