Distinct prime factors

(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).