If x is an integer such that x > 0, is it true that √x is an integer? (1) 9 × √(16x) is an integer. (2) √(6x) is not an integer.
A. Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not. B. Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not. C. Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, even though NEITHER statement BY ITSELF is sufficient. D. Either statement BY ITSELF is sufficient to answer the question. E. Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question, meaning that further information would be needed to answer the question.
(A) 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 wellknown 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).  I didn't understand the explanation of the answer, because in statement (1) I can write x = 1/4 and the answer will still be an integer.
