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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.

A very basic question can fractions be classified as even / odd. Sry if it is very basic. Got a sudden doubt in the test.

No, only integers can be even or odd. Even number is an integer which is divisible by 2 without a remainder and an odd number is an integer which when divided by 2 gives the remainder of 1.
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I don't go with the explanation provided for the question. As per the question , x is a positive integer . But, the explanation failed to consider this condition. For all positive integers(except 1) , the value of a is positive.