If x > 1, what is the value of integer x?
(1) There are x unique factors of x.
(2) The sum of x and any prime number larger than x is odd.

Statement 1: There are x unique factors of x.
The only value that works here is x=2: 2 has two unique factors (1 and 2), so the number of unique factors (2) is equal to x (2).
No other value works.
To illustrate:
If x=10, x will have 4 distinct factors (1,2,5,10), so the number of unique factors (4) is not equal to x (10).
If x=16, x will have 5 unique factors (1,2,4,8,16), so the number of unique factors (5) is not equal to x (16).
Thus, x=2.
SUFFICIENT.

Statement 2: The sum of x and any prime number larger than x is odd.
Any prime number greater than 2 is odd.
It's possible that x=2, since 2+odd = odd.
It's possible that x=4, since 4+odd = odd.
INSUFFICIENT.

The correct answer is A.

Hi GMATGuruNY ,

I really don't understand the question. Is the question is asking for exact value of x? If yes, then statement 1 shows that x has different value.

Please clear me sir.

Many thanks in advance.

SJ

jain2016 wrote:Hi GMATGuruNY ,

I really don't understand the question. Is the question is asking for exact value of x? If yes, then statement 1 shows that x has different value.

Please clear me sir.

Many thanks in advance.

SJ

Statement 1 is sufficient because it constrains the value of x to one possible number. There is only possible value of x, such that x has x unique factors - namely, 2. Two has two unique factors - 1 & 2.

jain2016 wrote:Hi GMATGuruNY ,

I really don't understand the question. Is the question is asking for exact value of x? If yes, then statement 1 shows that x has different value.

Please clear me sir.

Many thanks in advance.

SJ

Statement 1: There are x unique factors of x
If x=2, the statement above becomes:
There are 2 unique factors of 2.
This works, since 2 has two unique factors: 1 and 2.

If x=3, the statement above becomes:
There are 3 unique factors of 3.
This does NOT work, since 3 has two unique factors: 1 and 3.
Thus, x≠3.

If x=4, the statement above becomes:
There are 4 unique factors of 4.
This does NOT work, since 4 has three unique factors: 1, 2 and 4.
Thus, x≠4.

The ONLY value that works in the statement above is 2.
Thus, x=2.
SUFFICIENT.

Statement 1: There are x unique factors of x
If x=2, the statement above becomes:
There are 2 unique factors of 2.
This works, since 2 has two unique factors: 1 and 2.

If x=3, the statement above becomes:
There are 3 unique factors of 3.
This does NOT work, since 3 has two unique factors: 1 and 3.
Thus, x≠3.

If x=4, the statement above becomes:
There are 4 unique factors of 4.
This does NOT work, since 4 has three unique factors: 1, 2 and 4.
Thus, x≠4.

The ONLY value that works in the statement above is 2.
Thus, x=2.
SUFFICIENT.

[/quote]

Hi GMATGuruNY ,

Got it sir.

Thanks,

SJ

We might want to get into a little more detail about why S1 works.

If there are x unique factors of x, then x must be divisible by EVERY positive integer less than itself.

But for any positive integer n, n and (n + 1) will not share any prime factors, so (n + 1) will not be divisible by n UNLESS n = 1.

So the only positive integer that divides by the positive integer immediately before it is 2, meaning 2 is the only number that divisible by all positive integers less than or equal to itself.