Prime divisors

[GmatKiss]

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

1. 2N has one prime divisor
2. 3N has one prime divisor

[neelgandham]

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

1. 2N has one prime divisor

For 2N to have one prime divisor, N must be either 1(If N = 1, 2N = 2 and 2N has one prime divisor) or an integer of the form 2^k, where k is a positive integer(If N = 2^k, 2N = 2^(k+1) and 2N has one prime divisor , i.e. 2).
If N = 1, N has no prime divisors
If N = 2^k, N has one prime divisors, i.e. 2.
Since we have two different answers, statement 1 is insufficient to answer the question.

2. 3N has one prime divisor

For 3N to have one prime divisor, N must be either 1(If N = 1, 3N = 3 and 3N has one prime divisor) or an integer of the form 3^k, where k is a positive integer(If N = 3^k, 3N = 3^(k+1) and 3N has one prime divisor , i.e. 3).
If N = 1, N has no prime divisors
If N = 3^k, N has one prime divisors, i.e. 3.
Since we have two different answers, statement 2 is insufficient to answer the question.

From 1 and 2

The value of N is 1 and it has no prime divisors.
IMO C

[Anurag@Gurome]

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

1. 2N has one prime divisor
2. 3N has one prime divisor

(1) Since 2N has only one prime divisor, so 2N = 2, 4, 8, 16, 32.... This means N can take the values 1, 2, 4, 8, 16....
When N = 1, 2N = 2, which will have one prime divisor, 2 but N will not have any prime divisor.
When N = 2, 2N = 4, which again has 1 prime divisor 2, but in this case N will also have 1 prime divisor.
We don't get a unique answer.
So, (1) is NOT SUFFICIENT to answer the question.

(2) Since 3N has only one prime divisor, so 3N = 3, 9, 27, 81.... This means N can take the values 1, 3, 9, 27....
When N = 1, 3N = 3, which will have one prime divisor, 3 but N will not have any prime divisor.
When N = 3, 3N = 9, which again has 1 prime divisor 3, but in this case N will also have 1 prime divisor.
We don't get a unique answer.
So, (2) is NOT SUFFICIENT to answer the question.

Combining (1) and (2), we get that N = 1 is the only possible value, and 1 does not have any distinct prime divisor. Hence, N does not have any distinct prime divisors.

So, the correct answer is C.