How many different prime numbers are factors of the positive integer

How many different prime numbers are factors of the positive integer \(n\) ?

(1) \(2n\) has one prime factor.

Clearly, the sole prime factor of \(2n\) is 2. Thus, possible values for \(2n\) include 2, 4, 8, etc., implying \(n\) could be 1, 2, 4, and so forth. If \(n=1\), it doesn't have any prime factor, but if \(n\) takes any other value (2, 4, ...), it has one prime factor: 2. Not sufficient.

(2) \(3n\) has one prime factor.

The situation here mirrors the first: the only prime factor of \(3n\) is 3. Hence, \(3n\) could be 3, 9, 27, etc., which implies \(n\) could be 1, 3, 9, and so on. If \(n=1\), then it doesn't have a prime factor, but if \(n\) takes any other value (3, 9, ...), it has one prime factor: 3. Not sufficient.

(1)+(2) From above, the only possible value of \(n\) is 1, and 1 does not have any prime factors. Sufficient.


Answer: C