If x is a positive integer, which of the following CANNOT be expressed

Question

If x is a positive integer, which of the following CANNOT be expressed as n^2, where n is an integer?

A.  x^5

B.  x^2 − 1

C.  \sqrt{x^8}

D.  x^2 + 1

E.  \sqrt{x^5}

Bunuel · Accepted Answer

The question basically asks: if x is a positive integer, which of the following CANNOT be a perfect square.

Now, if x=1, then options A, B, C and E ARE perfect squares, therefore by POE the correct answer must be D.

Answer: D.

KarishmaB · Answer

Any random value of x will not help you get the answer. Even if you do not try x = 1, you can use reasoning to solve this question.

A.  x^5 
If x is a number with an even power, such as  x = a^4  (a is an integer), then  x^5 = a^{20} = n^2 
n will be  a^{10} , an integer here.

B.  x^2 - 1 
 x^2 - 1 = n^2 
You need two consecutive perfect squares. Only 0 and 1 are consecutive perfect squares. Thereafter, the distance between perfect squares keeps increasing. x needs to be a positive integers so if x = 1, n = 0 (an integer)

C.  \sqrt{x^8} 
 \sqrt{x^8} = x^4 = n^2 
n will be  x^2 , an integer here.

D.  x^2 + 1 
x is a positive integer so it must be at least 1. After 1, there are no two consecutive integers. n cannot be an integer.

E.  \sqrt{x^5} 
If x is a number with an even power which is a multiple of 4, such as  x = a^4  (a is an integer), then  \sqrt{x^5} = \sqrt{a^{20}} = a^{10} = n^2 
n will be  a^5 , an integer here.

Answer (D)

PareshGmat · Answer

If we take x = 2 & calculate, we cant get the answers;
 seems that x has to be taken 1 to execute all the options

jlgdr · Answer

So I guess zero does count as a perfect square then

Cheers
J

HarveyS · Answer

Even though Bunuel and karishma have answered this question, I am not able to digest any fundamental of this

.... Not sure how to crack this one... as I got this question in my Veritas prep exam today...

KarishmaB · Answer

Look, the question simply asks which option CANNOT be a perfect square. In the options, x is a positive integer.

Can x^5 be a perfect square? Can x take some value such that x^5 is a perfect square? Say, x = 4. Then x^5 = 4^5 = 2^10
This is a perfect square. Hence for some value of x, x^5 could be a perfect square. Hence this is not our answer. 
How do we find a value for which x^5 will be a perfect square? Perfect squares have even powers. We have x^5 which is an odd power. To get an even power, we could select x such that it already has an even power - we selected x = 2^2. Similarly, x could be 1^2 or 3^2 or 4^2 or 5^2 or 3^4 etc

Now think, can x^2 - 1 be a perfect square?
The reasoning for all the options is given in the post above.

A more intuitive approach is putting x = 1 as given by Bunuel. When x = 1, all options except option (D) results in a perfect square. So we know that all options CAN be perfect squares except (D). By elimination, answer must be (D).

dbiersdo · Answer

So the one point where I get tripped up is the X^2 - 1. 0 is assumed to be a perfect square?

How can I tell from the verbage of the question that what they are asking for is determining whether or not something is a perfect square or not?

Thanks for the help.

KarishmaB · Answer

Yes, both 0 and 1 are perfect squares.

"Can you express 'this' as n^2 where n is an integer?" asks us whether we can write 'this' as square of an integer. 
Square of an integer is a perfect square. So the question becomes "Can you express 'this' as a perfect square?"

Slightly convoluted verbiage is common in GMAT.

musunna · Answer

Please refer to option   where value comes 0.
Would you please clarify whether 0 is a perfect square?

Bunuel · Answer

Yes, 0 is a perfect square 0 = 0^2.

mvictor · Answer

I picked B, because I thought that 2 consecutive integers multiplied do not equal a perfect square...I did not take into consideration the possibility of 0...

Dirky · Answer

I think the hardest part about this question is working out what is being asked.

Which of the following cannot be expressed as n^2 means which of the following cannot be perfect square when n is an integer.

Subbing in 1 for x in each example brings us to answer D.

1^2 + 1 = 2, which is not the square of any integer. D is your answer! Easy

aventus · Answer

D.

x^2+1 = n^2 => x^2 = (n+1)(n-1) and n>0. This can never give a perfect square

dcummins · Answer

Official explanation:

D. For each of choices A, B, C, and E, trying the value x = 1 shows that there is one possible value of x for which the expression results in an integer squared:

A) The result is 1, which is the same thing as 1^2
B) The result is 0, which is the same thing as 0^2
C) The result is 1, which is the same thing as 1^2
E) The results is 1, which is the same thing as 1^2
For D, however, you cannot find a value of x for which x2+1
 will yield a perfect square. If x = 1, the expression yields a 2. If x = 2, the expression yields a 5. As no two squares (other than 0 and 1) are one number apart, D cannot be a perfect square if x is a positive integer.

Basshead · Answer

A. If  x = 4 ,  x^5  can be expressed as a perfect square.

B.  x^2 - 1 = n^2 
x must be equal to 1.

C.  x^4 = n^2 
 x^2 = n

D.  x^2 + 1  
If  x = 1 , then  x^2 + 1 = 2 . Not a perfect square.

E.  \sqrt{x^5}  =  \sqrt{1^5}  = 1

WheatyPie · Answer

It's not necessary that x be 1, that's just the easiest option.

In answer choice a, we could let  x=n^\frac{2}{5} , for example.

AbhishekP220108 · Answer

Since the ques ask for the cannot, which requires us to find which can't be a possible values for n2. As X is positive integer so taking x=1 gives us that D cant be a perfect square for n as an integer

If x is a positive integer, which of the following CANNOT be expressed

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GMAT Problem Solving (PS) Questions

If x is a positive integer, which of the following CANNOT be expressed

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