Data Sufficiency

ProGMAT wrote: If x is a positive integer, is sqrt(x) an integer?
(1) sqrt(4x) is an integer.
(2) sqrt(3x) is not an integer.

Target question: Is sqrt(x) an integer?

Given: x is a positive integer

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

So, if sqrt(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 sqrt(x) must be an integer.
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: sqrt(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 sqrt(3x) = sqrt(12), which is not an integer. In this case, sqrt(x) is an integer.
Case b: x = 5. This means that sqrt(3x) = sqrt(15), which is not an integer. In this case, sqrt(x) is not an integer.
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer = A

Cheers,
Brent

Hi fcabanski,

I started going that route as well, but then realized that, if 2sqrt(x) is an integer, then sqrt(x) is either an integer, or it equals something.5
For example, if sqrt(x) = 2.5, then 2sqrt(x) is an integer.
Proving that sqrt(x) cannot equal something.5 was a pain, so I went the prime factorization route.

Cheers,
Brent

ProGMAT wrote:

Brent@GMATPrepNow wrote: Target question: Is sqrt(x) an integer?

Given: x is a positive integer

Answer = A

Cheers,
Brent

Can we do it like:
Target question: Is sqrt(x) an integer?

Statement 1:sqrt(4x) is an integer.
If sqrt(4x) is an integer then 2*sqrt(x) will be an integer. As x will always be a sq of an integer to make 4x a square of something.
SUFFICIENT

Statement 2:sqrt(3x) not an integer.
sqrt(3) * sqrt(x) = Decimal * (Integer/Non-Integer)
It can be integer or non integer.
Can't Say. NOT SUFFICIENT

Hence A

Your rationale for statement 1 isn't entirely rigorous/thorough, but I think it'll do

Cheers,
Brent