If x is a positive number and a=xxx,

Question

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

Bunuel · Accepted Answer

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.

KarishmaB · Answer

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)

23a2012 · Answer

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?

EMPOWERgmatRichC · Answer

HI All,

The "font" (and lack of parentheses) makes this question a little tough to interpret. I believe that the intent is the following:

If X is POSITIVE, and A = (\sqrt{X})(X) - X which of the following MUST be true?

Since the question asks what MUST be true, we can quickly eliminate possibilities by proving what is NOT always true (by TESTing VALUES)

IF....
A = 1, then (\sqrt{1})(1) - 1 = (1)(1) - 1 = 0
This result is NOT positive. 
Eliminate Roman Numeral II and Answers B and D.

IF....
A = 2, then (\sqrt{2})(2) - 2 = NOT an integer.
Eliminate Roman Numerals I and III and Answers A and C.

Final Answer:  E

GMAT assassins aren't born, they're made,
Rich

KarishmaB · Answer

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.

ShashwatJha · Answer

We can aslo do the following. Just juggling of steps.

Given, a= sqrt(x)*x-x => a= x(sqrt (x) -1)
Then we start put to numbers for eg. x=1 => a=0. Cuts point 2
Put x>1, a may not be an integer and hence not even. Cuts point 1 and 3.

Ans is E.

gmatophobia · Answer

The given question is a   MUST   be true type of question , hence the option must hold true for all values.

We don't have enough information to determine if  \sqrt{x}  is an integer. Also its value is unknown.

Hence we cannot conclude on the nature of a.

Option E

Pragya9s · Answer

Must Be True question, hence should satisfy all the conditions.

a = x.sq root of x - x
= x ( Sq. root of x - 1)

Bring it to solvable equation,

1. X= 4 
= 4( 2 -1) 
= 4

2. X= 2 
= 2(Sq. Root of 2 - 1)
Not an Integer.

3. For Equation Not to be a Integer, Positive or Even

X = 1/4
= 1/4(1/2 -1)
= 1/4(-1/2)
= - 1/8

Hence , option E does not satisfy any option.

Akshaynandurkar · Answer

Statements 1 and 3 are easy to eliminate for second statement use x = 1 (positive number)
we get a = 0 which is not a positive number . statement 2 is also eliminated 
hence E

If x is a positive number and a=xxx,

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