If x is an integer, what is the sum of all distinct positive

Thanks!

(1) The only squares that have 3 distinct positive factors are 4, 25 and 49.
The factors are 1,x,\(\sqrt{x}\).
Since \(\sqrt{x}\) for the above three numbers needs to be considered, hence there will be three different values for the sums

Hence (1) is insufficient

(2) \(x^2-1 = 3k\) where k =odd integer

Hence, \(x^2 =3k+1\) is even

There can be multiple values where x^2 is even

Hence (2) is insufficient

(1)+(2) - x^2 should be even and should have 3 distinct positive factors
The only value that satisfies both conditions is 4

\(\sqrt{x}\) is 2, hence we can find the sum

Ans: C

I thought statement 2 was sufficient as \(\sqrt{x}\) is supposed to be an integer as only integer can have factors .
Based on this, statement 2 says product of 2 consecutive odd integers is odd
(x-1)(x+1)= odd

1*3= 3 here x= 2 but \(\sqrt{2}\) is not an integer hence we cannot take x= 2

3*5 = 15 here x= 4 and \(\sqrt{4}\) is an integer hence we can take x= 4

5*7 = cannot take, as this is not in the form 3K

7*9=63 here x = 8 but \(\sqrt{8}\) is not an integer hence we cannot take x= 8

9*11=99 X=10 but \(\sqrt{10}\) is not an integer hence we cannot take x= 10

so from statement 2 after testing various numbers I felt that x = 4 was the only value that qualified hence B was the answer.
Are there any values that I missed ?
or \(\sqrt{x}\) need not be an integer?

Factors are not essentially Integers otherwise the expressions of "HCF (Highest Common Factor) of Fractions" would not have existed.

In order to ensure that x is a perfect square we will have to combine the information of first statement with second statement. I hope this will give you an insight why answer option C is the correct answer.

well if non integers can have factors then the answer is C , Karishma if you can add little more to this that would surely help.

Here is another value: \(x^2 = 3*85 + 1 = 256\) --> \(x = 16\) --> \(\sqrt{x}=4\).

That's not true for the GMAT. Only positive integers are considered as factors and only integers could have factors on the GMAT.

Thanks for clearing that, should have tested few more values

Few more values that qualify for B
15*17 = 3.K ( for some positive integer K)
here x= 16 and \(\sqrt {16} =\) 4 sum of factors =7

63.65= 3.K here x = 64, \(\sqrt {64}\) = 8

99.101= 3.k here X= 100, \(\sqrt {100}\) =10
etc .

hence B is insufficient

1+2
Only x= 4 qualifies

It is very hard to prove something by just testing numbers. You may not have tested the right numbers or you may not have tested enough numbers. To prove something, use of logic is required. To disprove, it is much easier to use numbers since you only have to find one example where it doesn't hold.

From statement 2, you have [fraction]x^2 = 3x + 1[/fraction]
You need an even perfect square which is 1 more than a multiple of 3. But since \(\sqrt{x}\) must be an integer too, we are looking for an even fourth power which is 1 more than a multiple of 3.

An even fourth power is \(16 ( = 2^4)\) which is 1 more than 15. \(\sqrt{x} = 2\) here.
Another even fourth power is \(256 ( = 2^8)\) which is 1 more than 255 (multiple of 3). \(\sqrt{x} = 4\) here.
Another even fourth power is \(4096 = 2^{12}\) which is 1 more than 4095 (multiple of 3). \(\sqrt{x} = 8\) here.

and so on...

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