Integer

[vinay1983]

How many more positive integral factors does 2n have than the positive integer n?

1. n is an odd number

2. n is an odd prime number

[Brent@GMATPrepNow]

How many more positive factors does 2n have than the positive integer n?

1. n is an odd number

2. n is an odd prime number

Note: I removed the word "integral" from the question, since "factor" and "divisor" both imply integral values.

Target question: How many more positive factors does 2n have than the positive integer n?

Statement 1: n is an odd number
There are several values of n that satisfy this condition. Here are two:
Case a: n = 5, in which case n has 2 positive factors (1,5) and 2n has 4 positive factors (1,2,5,10). So, 2n has 2 more factors than n has
Case b: n = 9, in which case n has 3 positive factors (1,3,9) and 2n has 6 positive factors (1,2,3,6,9,18). So, 2n has 3 more factors than n has
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: n is an odd prime number
Since n is prime, we know that n will have only 2 positive factors (1 and n).
What about 2n?
The factors of 2n will be 1, 2, n and 2n for a total of 4 positive factors
So, 2n must have 2 more factors than n has
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer = B

Cheers,
Brent

[GMATGuruNY]

How many more positive integral factors does 2n have than the positive integer n?

1. n is an odd number

2. n is an odd prime number

Statement 1:
If n=1 (which has only 1 factor), then 2n = 2 (which has factors 1 and 2, for a total of 2 factors).
In this case, the difference between the number of factors = 2-1 = 1.
If n=3 (which has factors 1 and 3, for a total of 2 factors), then 2n=6 (which has factors 1, 2, 3, and 6, for a total of 4 factors).
In this case, the difference between the number of factors = 4-2 = 2.
INSUFFICIENT.

Statement 2:
If n=3 (which has factors 1 and 3, for a total of 2 factors), then 2n=6 (which has factors 1, 2, 3, and 6, for a total of 4 factors).
In this case, the difference between the number of factors = 4-2 = 2.
If n=5 (which has factors 1 and 5, for a total of 2 factors), then 2n=10 (which has factors 1, 2, 5, and 10, for a total of 4 factors).
In this case, the difference between the number of factors = 4-2 = 2.
The cases above illustrate that the difference in every case will be 2.
SUFFICIENT.

The correct answer is B.