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?

by VJesus12 » Fri Jun 11, 2021 7:51 am

00:00

A

B

C

D

E

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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.

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Re: How many distinct prime divisors does a positive integer $$n$$ have?

by Ian Stewart » Sat Jun 12, 2021 3:59 pm

00:00

A

B

C

D

E

Global Stats

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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