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 Post subject: GMAT Number Theory
PostPosted: Sat May 04, 2013 4:54 am 
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Is √(56x) an integer?
(1) x is a multiple of 14.
(2) 28 is not a factor of x.

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.


(E) We could pick numbers here, but it seems impractical given that, after multiplying, the numbers would be so large that it would be difficult to tell whether they were perfect squares.

Let’s try to simplify the question stem by taking the prime factorization of 56:
√(56x) = √(2 × 2 × 2 × 7x)
= √(2 × 2) √(2 × 7x)
= 2√(14x).

So, the question is really asking whether 2√(14x) is an integer.

Statement (1) is insufficient. There are certain multiples of 14 (like 14 or 56) that would make 2√(14x) an integer. In fact, the multiples that will work here are the cases where x equals 14 times a perfect square. There are other multiples of 14, like 28, that would not make it an integer.

Statement (2) is also insufficient. Most values of x that do not have 28 as a factor, such as 5, won't make 2√(14x) an integer, but there are some that will: 28 is not a factor of 14, yet setting x equal to 14 will make 2√(14x) an integer. So Statement (2) is also insufficient.

Combined, the two statements are still insufficient. As shown, x could be 14, and 2√(14x) would be an integer. But if x were 42, both statements would be satisfied, but 2√(14x) would not be an integer.

Since both statements are insufficient, even when combined, the correct answer is choice (E).
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So if both conditions are true then we can conclude that √(56x) is not an integer, correct? Shouldn't answer be C?

e.g. :
x = 42
(1) and (2) satisfied hence √(56) not an integer

However x = 28 (1) true but not (2) then we cannot say √(56x) is an integer or not


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 Post subject: Re: GMAT Number Theory
PostPosted: Sat May 04, 2013 4:55 am 
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Joined: Fri Apr 09, 2010 2:11 pm
Posts: 459
Quote:
So if both conditions are true then we can conclude that √(56x) is not an integer, correct?
No. For example these two values fit the both statements:

x = 14
(1) x = 14 is a multiple of 14
(2) 28 is not a factor of x = 14

and x = 42
(1) x = 42 is a multiple of 14
(2) 28 is not a factor of x = 42

And these two values yield different answers to the main question "Is √(56x) an integer?"
IF x = 14, then √(56x) = 28 . It is an integer!
√(56 × 14) = √(14 × 4 × 14) = 14 × 2 = 28

IF x = 42, then √(56x) = 28√3 . It is NOT an integer!
√(56 × 42) = √(14 × 4 × 14 × 3) = 14 × 2 × √3 = 28√3

So the both statements combined are NOT sufficient to give a definite answer to the main question.


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