If x is an integer....

we can simplify to
sqrt(x^3) - x = a

If x is negative, we cannot find its root, so we infer that x is not negative i.e. 0 or positive

to find which are true, we should attempt to disprove each of the choices

take x = 3 -> a = sqrt(27) - 3, which gives us a non-integer, so II and III are not always true and thus eliminiated

which leaves us with only answer choices B and E

now, by subsituting x = 0 -> a = sqrt(0) - 0 = 0, 0 is not positive and this B is eliminated

Therefore the answer choice is E.

Please correct me if am wrong.

joyseychow wrote: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


[spoiler]IMO is C. If I simplify equation sqrt(x) * x - x = a -----> x[sqrt(x) -1]=a, then am I right to say that
x=a or [sqrt(x) -1]=a. Hence shouldn't a be an integer since a=x??? I'm confuse...

OA is E.[/spoiler]

usually none of these is not a GMAT option.....
however here its easy to decipher that ans is E by POE
since nothing is mentioned about a
and abt x its an int , we dnt knw whether its +ve or -ve and we dnt anything about whether its even or odd

hence to compute anything abt a is impossible....
and to ans your query if
a*b=24
then to say either a=24 or b=24 is incorrect bcoz 8*3=24;6*4=24;12*2=24;24*1=24
but if a*b=0 then we can say either a=0 or b=0(its a property of 0)

here E is the ans

joyseychow wrote: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


[spoiler]IMO is C. If I simplify equation sqrt(x) * x - x = a -----> x[sqrt(x) -1]=a, then am I right to say that
x=a or [sqrt(x) -1]=a. Hence shouldn't a be an integer since a=x??? I'm confuse...

OA is E.[/spoiler]

We have an integer x, such that x √x - x = a.

Take example-numbers with the intent of denying a statement in all the “must be true” kind of questions on GMAT.

I. We do not assign terms like even/odd with the irrationals, x = 2, 3, 5… leave a as a complex irrational, so NO.

II. It is certain that x is a non-negative integer, otherwise GMAT would not have put that under ‘√’. When x =1, a = 0; sufficient reason to deny II.

III. An irrational is not an integer, see I, and deny III.

All denied

[spoiler]E[/spoiler]