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GMAT Diagnostic Test Question 3 Field: Arithmetic, Roots Difficulty: 700-750

Rating:

If \(x\) is an integer and \(\sqrt{x} * x - x = a\) , 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

BELOW IS REVISED VERSION OF THIS QUESTION:

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.

agree for Not integer , +ve but how we can prove for EVEN ??

If a number is not an integer, then for sure is not even. The question asks "...must be true?" So, if the number can be a non-integer, it means that it can be not even.
_________________

PhD in Applied Mathematics Love GMAT Quant questions and running.

You cannot divide an equation by \(x,\) unless you know that \(x\) is non-zero. In the original question, \(x\) was given as an integer, so it can be 0. In the revised question (by Bunuel), \(x\) was given as a positive integer, therefore in this case, you are allowed to divide by \(x,\) because now you are sure that \(x\) cannot be 0.

It is not clear why from the equation \(\sqr{x}=1+\frac{a}{x}\) it is (more) evident that \(a\) and \(x\) could be any number.
_________________

PhD in Applied Mathematics Love GMAT Quant questions and running.

GMAT Diagnostic Test Question 3 Field: Arithmetic, Roots Difficulty: 700-750

Rating:

If \(x\) is an integer and \(\sqrt{x} * x - x = a\) , 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

The question asks for statements that must be true. There are infinitely many positive numbers. The question is whether \(a\) can be 0? The answer is YES! The smallest possible value for \(x\) is 1, for which \(a = 0.\) So, definitely, statement II must not necessarily be true.

Choosing \(x=2\), we obtain \(a=2(\sqr{2}-1),\) which evidently is not an integer. Not being an integer, for sure is not an even number. We can immediately deduce that neither III, nor I must be true.

Answer E.
_________________

PhD in Applied Mathematics Love GMAT Quant questions and running.

GMAT Diagnostic Test Question 3 Field: Arithmetic, Roots Difficulty: 700-750

Rating:

If \(x\) is an integer and \(\sqrt{x} * x - x = a\) , 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

BELOW IS REVISED VERSION OF THIS QUESTION:

If \(x\) is a positive integer 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.

\(x=\frac{1}{4}\) is not an integer.
_________________

PhD in Applied Mathematics Love GMAT Quant questions and running.

GMAT Diagnostic Test Question 3 Field: Arithmetic, Roots Difficulty: 700-750

Rating:

If \(x\) is an integer and \(\sqrt{x} * x - x = a\) , 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

BELOW IS REVISED VERSION OF THIS QUESTION:

If \(x\) is a positive integer 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.

\(x=\frac{1}{4}\) is not an integer.

There was a typo. It should be "if x is a positive number".
_________________

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.

How can you assume x = 1/4 when the question states that x is an integer?!
_________________

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.

How can you assume x = 1/4 when the question states that x is an integer?!

There was a typo. It should be "if x is a positive number".

If \(x\) is an number and \(\sqrt{x} * x - x = a\) , 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[/quote]

This question we make assumptions by taking the value of x as follows:-

x * sqrt(x) - x = a.

if we choose x=0, so lets compute the calculation of it.

a) If x=0, we have a=0. ie 0 is neither negative nor positive number.

b) If x=4, 4 * 2 - 4 = 8-4 = 4.

c) If x=2, we obtain a=2(\sqr{2}-1), which evidently is not an integer. Not being an integer, for sure is not an even number. We can immediately deduce that neither III, nor I must be true.