pavithra14 wrote:Is the square root of the positive integer X an integer?

(1) The sum of the distinct factors of X is odd.

(2) X has an odd number of distinct factors.

This question is improperly worded, and could never be a true GMAT question (without some editing).
I say this because it's impossible for an integer to have an odd number of distinct factors. It's also impossible for the sum of the factors to be odd.

For example, the factors (divisors) of 9 are {1, 3, 9, -1, -3, -9}
Similarly, the factors (divisors) of 6 are {1, 2, 3, 6, -1, -2, -3, -6}
For every positive factor of X, there's also a negative factor of X. So, the sum of the factors will ALWAYS equal zero (which is not odd), and there will always be an EVEN number of factors.

This question SHOULD read:

Is the square root of the positive integer X an integer?
(1) The sum of the distinct positive factors of X is odd.
(2) X has an odd number of distinct positive factors.

Cheers,
Brent

Let's solve this GMAT-style question

Is the square root of the positive integer X an integer?

(1) The sum of the distinct positive factors of X is odd.
(2) X has an odd number of distinct positive factors.

Target question: Is the square root of the positive integer X an integer?
This question is a great candidate for rephrasing the target question.

Aside: We have a free video with tips on rephrasing the target question: https://www.gmatprepnow.com/module/gmat- ... cy?id=1100

If the square root of X is an integer, what must be true about X?
X must be the square of an integer (e.g., 1, 4, 9, 16, etc.)

Rephrased target question: Is X the square of an integer?

Statement 1: The sum of the distinct positive factors of X is odd.
There are many values that meet this condition. Here are two:
Case a: X = 1 (1 has only 1 as its factor, so the sum = 1, which is odd). In this case X is the square of an integer
Case b: X = 2 (the factors of 2 are 1 and 2, so the sum = 3, which is odd). In this case X is not the square of an integer
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: X has an odd number of distinct positive factors.
There's a nice rule that says, "If positive integer k has an odd number of distinct factors, then k is the square of an integer"
Some examples: The factors of 9 are {1,3,9}. There is an odd number of factors and 9 is a square
The factors of 36 are {1,2,3,4,6,9,12,18,36}. There is an odd number of factors and 36 is a square
The factor of 1 is {1} There is an odd number of factors and 1 is a square
Conversely, the factors of 6 are {1,2,3,6}. There is an even number of factors and 6 is not a square
The factors of 10 are {1,2,5,10}. There is an even number of factors and 10 is not a square
The factors of 20 are {1,2,4,5,10,20}. There is an even number of factors and 20 is not a square
So, given the above property, we can be certain that X is the square of an integer
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer = B

Cheers,
Brent