How many distinct prime divisors does a positive integer \(n\) have?

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How many distinct prime divisors does a positive integer \(n\) have?

(1) \(2n\) has one distinct prime divisor.

(2) \(3n\) has one distinct prime divisor.

Answer: C

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If n is a positive integer, 2n is clearly divisible by 2. If, as Statement 1 says, 2n has only one prime divisor, that prime divisor must be 2. But then n can be 1, 2, 2^2, 2^3 or any other power of 2. Since it is possible n = 1, it is possible n has no prime divisors, and if n is any other power of 2, then n has one prime divisor.

Similarly Statement 2 means n is 1, 3, 3^2, 3^3 or any other power of 3. So n might have no prime divisors or one prime divisor.

Combining the Statements, the only possibility is that n = 1, so the answer is C.
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