Distinct prime divisors

[rahulvsd]

How many distinct prime divisors does a positive integer N have?

A. 2N has one prime divisor
B. 3N has one prime divisor

[spoiler]OA: C. [/spoiler]

[aneesh.kg]

One of the factors of 2N is 2, which is prime. But if 2N has just one prime divisor, N cannot be prime and has to be of the form 2^n. (If N is has any other prime factor but 2, 2N will also have that prime factor). So, N has (n+1) factors and we don't know a specific value.
Statement (1) is insufficient.

Similar logic holds for the Statement(2). If 3 is one of the factors, then N cannot be prime and has to be of the form 3^n(If N is has any other prime factor but 3, 3N will also have that prime factor). So, N has (n+1) factors and we don't know a specific value.
Statement (2) is also insufficient.

Lets try to combine the two statements. If N is of the form 2^n and 3^n, then n = 0 and N has to be 1.
Thus N has just one positive factor and the answer is (C).

[GMATGuruNY]

How many distinct prime divisors does a positive integer N have?

A. 2N has one prime divisor
B. 3N has one prime divisor

[spoiler]OA: C. [/spoiler]

Statement 1: 2N has one prime divisor
2 is a prime divisor of 2N.
For 2N to have no OTHER distinct prime divisors, N = 2^x.
If N = 2^0 = 1, then it has NO prime divisors.
If N = 2^x (where x>0), then it has ONE distinct prime divisor: the number 2 itself.
INSUFFICIENT.

Statement 2: 3N has one prime divisor
3 is a prime divisor of 3N.
For 3N to have no OTHER distinct prime divisors, N = 3^x.
If N = 3^0 = 1, then it has NO prime divisors.
If N = 3^x (where x>0), then it has ONE distinct prime divisor: the number 3 itself.
INSUFFICIENT.

Statements 1 and 2 combined:
The only value that satisfies both statements is N=1, in which case N has NO distinct prime divisors.
SUFFICIENT.

The correct answer is C.