If x is a positive integer, is \sqrt{x} an integer?

[pims]

If x is a positive integer, is \sqrt{x} an integer?

(1) sqrt(4x) is an integer.

(2) sqrt(3x) is not an integer.


OA:A

[pims]

Why is the OA A?

2*sqrt(x)=INT

2*sqrt(x)=1 --> sqrt(x)=0.5
2*sqrt(x)=2 -- sqrt(x)=1 ??

[GMATGuruNY]

If x is a positive integer, is √x an integer?

(1) √(4x) is an integer.

(2) √(3x) is not an integer

Statement 1: √(4x) is an integer
Since x must be a positive integer, the smallest possible value for x is 1, with the result that √(4x) = √(4*1) = 2.
If x=1, then √x = √1 = 1.

The next greatest possible integer value for x is 4, with the result that √(4x) = √(4*4) = 4.
If x=4, then √x = √4 = 2.

The next greatest possible integer value for x is 9, with the result that √(4x) = √(4*9) = 6.
If x=9, then √x = √9 = 3.

In every case, √x is an INTEGER.
SUFFICIENT.

Statement 2: √(3x) is not an integer
Case 1: x=1, with the result that √(3x) = √3 = non-integer.
In this case, √x = √1 = 1, which is an integer.
Case 2: x=2, with the result that √(3x) = √6 = non-integer.
In this case, √x = √2, which is NOT an integer.
Since √x is an integer in Case 1 but a non-integer in Case 2, INSUFFICIENT.

The correct answer is A.

Why is the OA A?

2*sqrt(x)=INT

2*sqrt(x)=1 --> sqrt(x)=0.5
2*sqrt(x)=2 -- sqrt(x)=1 ??

The case in red violates that constraint that x must be a POSITIVE INTEGER.

[Brent@GMATPrepNow]

If x is a positive integer, is √x an integer?

(1) √(4x) is an integer
(2) √(3x) is not an integer

Here's another approach:

Target question: Is √x an integer?

Given: x is a positive integer

Statement 1: √(4x) is an integer
IMPORTANT CONCEPT: If K is an integer, then √K will be an integer if the prime factorization of K has an even number of each prime.
Some examples:
√144 = 12 (integer), and 144 = (2)(2)(2)(2)(3)(3) [four 2's and two 3's]
√1600 = 40 (integer), and 1600 = (2)(2)(2)(2)(2)(2)(5)(5) [six 2's and two 5's]
√441 = 21 (integer), and 441 = (3)(3)(7)(7) [two 3's and two 7's]
√12 = some non-integer, and 12 = (2)(2)(3) [two 2's and one 3's]

So, if √(4x) is an integer, then the prime factorization of 4x has an even number of each prime.
Since 4x = (2)(2)(x) we can see that the prime factorization of x must have an even number of each prime.
If the prime factorization of x has an EVEN number of each prime, then √x must be an integer.
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: √(3x) is not an integer.
There are several values of x that meet this condition. Here are two:
Case a: x = 4. This means that √(3x) = √12, which is not an integer. In this case, √x is an integer.
Case b: x = 5. This means that √(3x) = √15, which is not an integer. In this case, √x is not an integer.
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer = A

Cheers,
Brent