If x > 1, what is the value of integer x?

First of all, it is not a very straight forward question. Definitely needs some thinking so relax...

Ques: What is the value of x?
So we are looking for a single value of x.

First consider statement 2 since it is easier.
x + prime number greater than x = odd
There will be many many prime numbers greater than x. All prime numbers are odd except 2. So if you can add any prime number greater than x to x and get an odd number, it means x must be even. (because Even + Odd = Odd)
So all statement 2 tells you is that x is even. It could be 2 or 4 or 6 etc

Now look at statement 1.
There are x unique factors of x. Think of the first number greater than 1.
2 has 2 unique factors: 1 and 2 ( 2 is a prime number)
What about 3? It has 2 unique factors: 1 and 3 (a prime number)
4 has 3 unique factors: 1, 2 , 4
Is it possible that any greater number x has x unique factors? No. Why?
For x to have x unique factors, each number from 1 to x must be a factor of x. Say if 10 had 10 unique factors, each number 1, 2, 3, 4, 5..., 9,10 would have to be a factor of 10 (because factors are always positive integers)
But can 9 be a factor of 10 i.e. can (x-1) be a factor of x? No. 2 consecutive positive integers share only one common factor i.e. 1. Why? Check out the post given below for the answer:
question-from-practice-exam-78880.html#p847817

So statement 1 is enough to tell us that x is 2.

Thanks everyone!! Really much appreciated, I get it now.. This has been extremely helpful.. I need to do a GMAT but it's been 6 years since I've studied maths.. And even in high school I was horrible so i'm really stressed about this.. Hopefully I'll get better haha.. Again, thanks!

They could have worded Statement 1 better, but it's just trying to tell you how many positive divisors x has. Here we learn that x has x positive divisors. If you just imagine a number for x (e.g. imagine the phrase "10 has 10 positive divisors", clearly absurd) you can probably see quickly that this is almost never true. For x to have x positive divisors, x would need to be divisible by *every* integer from 1 up to x inclusive. That will only happen if x = 1 or x = 2. Since x > 1, x must be 2, and Statement 1 is sufficient.

Statement 2 will be true for any even value of x (since any prime larger than x will be odd if x > 2), so is not helpful.

The answer is A.

IMO the answer has to be A

Option A says x unique factors of x ...
In my opinion there are only two integers whose unique factors are equal to themselves
They are 1 = 1 and 2 = 1,2
And We need an integer greater that 1 so it has to be 2
Thus sufficient

Option B tells us that x+prime num larger that x = odd
therefore 2 + 3 = 5 ...odd
also if we suppose x= 4 then 4 +5 = 9...odd ....

therefore x can be any even number greater than 1 ...hence this option is insufficient

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.

The MGMAT book explains the answer as 1) S, 2) NS. They say that in order for statement one to be true, every integer between 1 and x, inclusive, must be a factor of x. By testing numbers, this holds true for 1 and 2, but not for 3 and 4. (Page 31 of the number properties guide if anyone cares to look).

Can someone please explain to me why this does not hold true for the numbers 3 and 4. Not sure what I am missing here.

Tell if you are not clear.

the question asks what number is equal to the number of his unique factors. 1 has 1 unique factor so 1 is equal to the number of his unique factors. 2 is equal to the number of his unique factors.
3 has only 2 unique factors , 4 and 5 have 2, 11 only 2 and so on.

so the only numbers equal to their unique fators are 1 and 2 . the condition is that x>1 so only 2 applies.

Why are you guys not considering the number itself as a factor?

To me:

4 has 3 unique factors (not 2) - 1,2,4
5 has 2 unique factors - 1,5
6 has 4 (not 3) unique factors - 1,2,3,6

It doesn't change the answer or anything, but the number itself is a unique factor. They weren't asking for the prime roots.

Yes - This is correct.....I had missed this...

And why does this not affect the answer ?
Because
Total number of unique factors for any number = 1 + Total number of unique factors other than the number
Adding 1 does not affect the count because we are any way comparing the (actual count - 1)
However my solution will be wrong if applied for larger numbers say 12
\(12 = 3 *2^2\)
Number of unique factors other than \(1 = (1+1)*(2+1) - 1 = 2*3 - 1 = 5\)

You can also calculate and confirm this :
Factors of 12 - 1,2,3,4,6,12
Factors of 12 other than 1 = 2,3,4,6,12
All are unique
Total number of such unique factors other than 1 = 5


So - Substitute smartly because this is a DATA SUFFICIENCY and NOT PROBLEM SOLVING question.
I am editing my post to correct this ....

If x>1, what is the value of integer x?

(1) There are x unique factors of x --> x to have x distinct positive factors it must be divisible by EVERY integer from 1 to x, inclusive but x can not be divisible by x-1 unless x=2, for example 3 is not divisible by 2, 4 is not divisible by 3, etc, but 2 is divisible by 1 (other value of x which has x distinct positive factors (1) is excluded by the stem as x>1). So this statement implies that x=2. Sufficient.

(2) The sum of x and any prime number larger than x is odd --> x+prime=odd, now as x itself is more than 1 then prime more than x cannot be 2 thus it's odd and we have x+odd prime=odd --> x=even, so x could be ANY even number. Not sufficient.

Answer: A.

OPEN DISCUSSION OF THIS QUESTION IS HERE: if-x-1-what-is-the-value-of-integer-x-110661.html