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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

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.

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

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.

If x is a positive number and a=√x∗x−x, [#permalink]

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18 Jan 2015, 13:22

Bunuel wrote:

gmatgambler wrote:

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

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.

[b][color=#0d004c]I did as Bunuel did where I used 1/4 as option and I subtrac 1/4 from 1/4 and then Multiple by sqrt 1/4

the result was always zero. So my answer was E because 0 is both even and integer abd there is no option include both A and C. Does my reason to chosen was true?
_________________

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

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.

[b][color=#0d004c]I did as Bunuel did where I used 1/4 as option and I subtrac 1/4 from 1/4 and then Multiple by sqrt 1/4

the result was always zero. So my answer was E because 0 is both even and integer abd there is no option include both A and C. Does my reason to chosen was true?

Multiplication/Division always comes before Addition/Subtraction. You cannot subtract x first and then multiply by root(x). The multiplication has to be done first because sequence of operators follows the rule of PEMDAS/BODMAS. I suggest you to check this out from a Math book or online.
_________________

Re: If x is a positive number and a=√x∗x−x, [#permalink]

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30 Oct 2016, 07:22

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Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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