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If x represents the number of positive factors of integer y,

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by BTGmoderatorDC » Wed Nov 20, 2019 5:51 pm
If x represents the number of positive factors of integer y, is x odd?

(1) y = n! where n is a positive integer greater than 1
(2) y = m^2 − 1 where m is a positive integer greater than 1

OA D

Source: Veritas Prep
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Source: — Data Sufficiency |

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by Jay@ManhattanReview » Thu Nov 21, 2019 10:42 pm
BTGmoderatorDC wrote:If x represents the number of positive factors of integer y, is x odd?

(1) y = n! where n is a positive integer greater than 1
(2) y = m^2 − 1 where m is a positive integer greater than 1

OA D

Source: Veritas Prep
Note that the number of positive factors of a non-perfect square number is always even. Or, the number of positive factors of a perfect square number is odd. For example, the no. of factors of 4 is 3 (1, 2 and 4); the no. of factors of 9 is 3 (1, 3 and 9).

So, the question is whether y is a perfect square no.

Let's take each statement one by one.

(1) y = n! where n is a positive integer greater than 1.

Since n is a positive integer greater than 1, n! cannot be a perfect square. Thus, the number of positive factors of y is even. The answer is no. Sufficient.

(2) y = m^2 − 1 where m is a positive integer greater than 1.

We see that m^2 is a perfect square no; thus, m^2 - 1 is not. Thus, the number of positive factors of y is even. The answer is no. Sufficient.

The correct answer: D

Hope this helps!

-Jay
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