Hi rishianand7,

This type of DS question is great for TESTing values.

We're told that X is a positive integer and we're asked if the square root of X is an integer. This is a YES/NO question.

Fact 1: The sum of the DISTINCT factors of X is odd

If x = 1, then the only factor is 1 (which fits what we know) and the answer to the question is YES
If x = 2, then the distinct factors are 1, 2 (1 + 2 = 3, which fits) and the answer to the question is NO
Inconsistent = INSUFFICIENT

Fact 2: X has an odd number of distinct factors

If x = 1, then there's 1 factor (which fits) and the answer to the question is YES
If x = 4, then there are 3 factors (1, 2, 4, which fits) and the answer is YES
If x = 9, then there are 3 factors (1, 3, 9, which fits) and the answer is YES

X CAN'T be 2, 3, 5, 7, 8 because these numbers don't have an ODD number of distinct factors.
Consistent = SUFFICIENT

Final Answer: B

GMAT assassins aren't born, they're made,
Rich

rishianand7 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.

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 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 factors.
melguy mentions a useful property: "If X has an odd number of distinct factors, then X is the square of an integer" (NOTE: we'll examine this property at the end of the post).
Given this 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

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Now, let's take a closer look at melguy's useful property: "If X has an odd number of distinct factors, then X is the square of an integer"
Why is this?
Notice that, when we find factors, we can typically do so by finding pairs of values that have a some particular product.
So, for example, to find the factors of 12, we can observe that (1)(12)=12, (2)(6)=12, and (3)(4)=12. So, the factors of 12 are 1,2,3,4,6 and 12.
So, if we have several pairs of factors (like we have above), then we will get an EVEN number of factors.

Under what circumstances will we get an ODD number of factors?
This will occur when one pair of factors has IDENTICAL VALUES.
For example, if we use pairs of values to find the factors of 36, we get (1)(36)=36, (2)(18)=36, (3)(12)=36, (4)(9)=36, (6)(6)=36.
So, we have 4 pairs of DIFFERENT values, and 1 pair of IDENTICAL values. This gives us an ODD number of factors.
So, the factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, 36

Cheers,
Brent

rishianand7 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.

I should mention that 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