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16 Sep 2014, 01:05
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How many different prime numbers are factors of the positive integer \(n\) ?
(1) \(2n\) has one prime factor.
(2) \(3n\) has one prime factor.
(1) \(2n\) has one prime factor.
(2) \(3n\) has one prime factor.
16 Sep 2014, 01:05
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Official Solution:
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
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
yashrakhiani
Joined: 26 Jul 2016
Last visit: 10 Nov 2016
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Concentration: Operations, Entrepreneurship
Schools: Tepper '19
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09 Oct 2016, 11:16
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Bunuel
my doubt: we could interpret this from each statement alone that only 1 has no prime divisors & no other number satisfies this condition, then why go further?
