If x is a Positive Integer, is\(\sqrt{x}\) an Integer GMAT Data Sufficiency

Question: If x is a positive integer, is \(\sqrt{x}\) an integer?

1. \(\sqrt{7*x}\)
2. \(\sqrt{9*x}\)

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are NOT sufficient.

Answer:
Solution with Explanation:
Approach Solution (1):

If x is a positive integer then \(\sqrt{x}\) is either a positive integer or an irrational number. (It cannot be some reduced fraction e.g. \(\frac{7}{3} or \frac{1}{2}\)).
Also, note that the question basically asks whether x is a perfect square.

(1)\(\sqrt{7*x}\) is an integer. This statement tells that x cannot be a perfect square, because it is, for example if \(x=n^2\)for some positive integer n, then \(\sqrt{7x}=\sqrt{7n^2}=n\sqrt{7}\neq\) integer
Sufficient

(2)\(\sqrt{9*x}\) is not an integer

Simplify: \(\sqrt{9*x}=3*\sqrt{x}\neq\)integer, which means that \(\sqrt{x}\neq\) integer
Sufficient

Correct Option: D

Approach Solution (2):

S1: x is square root of an odd power of 7, so not an integer.
Sufficient

S2: x is not a perfect square; hence its square root will not be an integer
Sufficient

Correct Option: D

“If x is a positive integer, is\(\sqrt{x}\) an integer?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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