How many distinct prime divisors does a positive integer N have?
1. 2N has one prime divisor
2. 3N has one prime divisor
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).
