NUMBERS

If x is a positive integer, which of the following CANNOT be expressed as n², where n is an integer?

A) x�
B) x² - 1
C) √x�
D) x² + 1
E) √x�

n² is a PERFECT SQUARE.
Question stem, rephrased:
Which of the following is NOT a perfect square?
Perfect squares are 0, 1, 4, 9, 16, 25...

Test the SMALLEST POSSIBLE CASE.
Test x=1 in the answer choices.
Eliminate any answer choice that becomes a perfect square when x=1.

A) x� = 1� = 1.
B) x² - 1 = 1² - 1 = 1 -1 = 0.
C) √x� = √1� = √1 = 1.
D) x² + 1 = 1² + 1 = 1 + 1 = 2.
E) √x� = √1� = √1 = 1.

Eliminate the options in red.

The correct answer is D.

ash4gmat wrote:If x is a positive integer, which of the following CANNOT be expressed as n², where n is an integer?

In case you wouldn't have seen Mitch's elegant way of getting to the answer, here is how you could hack this question.

A) x� Make x a perfect square, such as 4. Since 4 itself is a perfect square, 4 to any power will be a perfect square.

B) x² - 1 This one looks tougher. Leave it for now, as it could be the answer.

C) √x� Obviously the square root of something to the 8th power will be a perfect square.

D) x² + 1 This one looks tougher. Leave it for now, as it could be the answer.

E) √x� Make x a perfect fourth power, such as 16, so that √x� is still a perfect square.

We are left with B and D, and need to find a difference.

Try some numbers.

B) 2² - 1 = 3, 3² - 1 = 8, 4² - 1 = 15 None working.

D) 2² + 1 = 5, 3² + 1 = 10, 4² + 1 = 17 None working.

There has to be a difference, a case where one of these works.

D seems as if it will never work. Is there a case in which B works? Seeing certain cases is often the key to getting GMAT quant answers.

There is a case in which B works.

1² - 1 = 0, a perfect square.

So we hacked our way to the correct answer, D.