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03 Feb 2014, 05:41
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E
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Difficulty:
55%
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Question Stats:
57% (01:31) correct
43%
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wrong
based on 866
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If x is a positive number and a=√x∗x−x, which of the following must be true?
I. a is even
II. a is positive
III. a is an integer
A)I only
B)II only
C)III only
D)I and II
E) None of the above
I. a is even
II. a is positive
III. a is an integer
A)I only
B)II only
C)III only
D)I and II
E) None of the above
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03 Feb 2014, 05:50
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gmatgambler
If \(x\) is a positive number and \(a=\sqrt{x} * x - x\), which of the following must be true?
I. \(a\) is even
II. \(a\) is positive
III. \(a\) is an integer
A. I only
B. II only
C. III only
D. I and II
E. None of the above
Note that we are asked "which of the following MUST be true, not COULD be true. For such kind of questions if you can prove that a statement is NOT true for one particular set of numbers, it will mean that this statement is not always true and hence not a correct answer.
If \(x=\frac{1}{4}\) then \(a=\sqrt{x} * x - x=\frac{1}{2}*\frac{1}{4}-\frac{1}{4}=-\frac{1}{8}\). Now, \(-\frac{1}{8}\) is not an integer at all (hence not even) and also not positive, so none of the options MUST be true.
Answer: E.
03 Feb 2014, 22:44
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gmatgambler
Also, let me point out how we arrive at the numbers that we should try here. Given x is a positive number and \(a =\sqrt{x}*x - x\).
\(\sqrt{x}\) makes us think of perfect squares. We try to put in a perfect square such as 4 and see that all conditions are met.
So now we need to think of the cases when the condition may not be met.
Let's figure out whether a must be positive.
\(\sqrt{x}\) is positive and x is positive but for a to be positive, \(\sqrt{x}*x\) MUST be greater than x. Will \(\sqrt{x}*x\) be less than x in any case? It will be if \(\sqrt{x}\) is less than 1 i.e. a fraction. So this leads us to x = 1/4. We must try this value. When x = 1/4, we see that a is not positive, is not an integer and hence the question of even or odd doesn't arise.
Answer (E)
