"The difference of ANY two positive factors of n is odd"

Remember, any two.

This means that if you were testing, say 12:

12 has the factors : 1, 2, 3, 4, 6, and 12

If you pick 1 and 3, or 3 and 12, you will get even differences. So (2) is violated.

Only an ( Even - Odd ) or ( Odd - Even ) can give you an Odd difference. This is an important number property for you to know.

This means that n has only 1 even and 1 odd factor, since if it had more than 1 of either of them you could pick 2 of the kind and get an even difference.

This is only true of the number 2 which has factors 1 and 2. Difference = 1

Hope that clears your doubt!

Hi kvcpk,

Is n=2?

1) exactly 2 factors --> n is prime. Could be 2, but could also be 3. NOT SUFFICIENT
2) Any two factors of n must have an odd difference:

• n cannot be odd because two of its factors (n and 1) would have an even difference.

• n cannot be an even greater than 2 because two if its factors (n and 2) would have an even difference.

• n must be 2, because the difference between its only 2 factors (2 and 1) would be odd.

n=2. SUFFICIENT
[spoiler]
The answer is B[/spoiler]. A more detailed solution as well as a step by step video solution can be found at GMATPrep question 1017. Use the Drill Engine to generate similar questions in a timed drill. Set topic='Number Properties' and difficulty='700+'

Patrick & albatross86

Thanks for the explanation. I understand it now. Was looking at the seond premise in a wrong way.

Praveen