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If a positive integer \(n\) has exactly two positive factors, what is the value of \(n\)?
(1) \(\frac{n}{2}\) is one of the factors of \(n\).
(2) The lowest common multiple of \(n\) and \(n + 10\) is an even number.
(1) \(\frac{n}{2}\) is one of the factors of \(n\).
(2) The lowest common multiple of \(n\) and \(n + 10\) is an even number.
16 Sep 2014, 01:44
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Bookmarks
Official Solution:
If a positive integer \(n\) has exactly two positive factors, what is the value of \(n\)?
Notice that, "\(n\) has exactly two positive factors" simply means that \(n\) is a prime number, so its factors are 1 and \(n\) itself.
(1) \(\frac{n}{2}\) is one of the factors of \(n\).
Since \(\frac{n}{2}\) cannot equal \(n\), then \(\frac{n}{2}=1\), thus \(n=2\). Sufficient.
(2) The lowest common multiple of \(n\) and \(n + 10\) is an even number.
If \(n\) is an odd prime, then \(n+10\) is also odd. The LCM of two odd numbers cannot be even, therefore \(n\) is an even prime, so \(n=2\). Sufficient.
Answer: D
If a positive integer \(n\) has exactly two positive factors, what is the value of \(n\)?
Notice that, "\(n\) has exactly two positive factors" simply means that \(n\) is a prime number, so its factors are 1 and \(n\) itself.
(1) \(\frac{n}{2}\) is one of the factors of \(n\).
Since \(\frac{n}{2}\) cannot equal \(n\), then \(\frac{n}{2}=1\), thus \(n=2\). Sufficient.
(2) The lowest common multiple of \(n\) and \(n + 10\) is an even number.
If \(n\) is an odd prime, then \(n+10\) is also odd. The LCM of two odd numbers cannot be even, therefore \(n\) is an even prime, so \(n=2\). Sufficient.
Answer: D
General Discussion
