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 Post subject: GMAT Algebra (Data Sufficiency)Posted: Tue Aug 17, 2010 11:00 am

Joined: Sun May 30, 2010 3:15 am
Posts: 424
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 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√54 is not an integer and √9 = 3, however, if x = 2, √12 is not an integer and neither is √2. So, Statement (2) is insufficient, and the answer choice is (A).
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Statement 1 must be insufficient. For example, if √x = 0.5, then statement 1 can still be true. Conversely, if the √x = (any integer), then statement 1 is still true.

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 Post subject: Re: math (test 5, question 28): number theory, data sufficiePosted: Tue Aug 17, 2010 11:10 am

Joined: Sun May 30, 2010 2:23 am
Posts: 498
Note, that x is a positive integer, thereofore it is not possible for √x to be 0.5. In that case x would equal 1/4.

The fact that x is a positive integer is highly important in this question. Let us analyse statement (1): 9 × √(16x) is an integer.

9 × √(16x) = 9 × 4 ×√x = a, where a is an integer
So √x = a / 36
x = (a / 36)²
Remember, that x is a positive integer, therefore a must be divisible by 36 and therefore √x = a / 36 is an integer as well.

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 Post subject: Re: math (test 5, question 28): number theory, data sufficiePosted: Mon Sep 06, 2010 12:45 pm

Joined: Sun May 30, 2010 3:15 am
Posts: 424
The right answer is E - statements are not sufficient, since x could be 1 or 1/9.

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 Post subject: Re: math (test 5, question 28): number theory, data sufficiePosted: Mon Sep 06, 2010 12:49 pm

Joined: Sun May 30, 2010 2:23 am
Posts: 498
Note, that question statement tell us: "..x is an integer..".
Therefore it is not possible for x to be 1/9.

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