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16 Sep 2014, 01:45
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If \(p\) is a positive integer, is \(p\) a prime number?
(1) \(p\) and \(p+1\) have the same number of positive factors.
(2) \(p-1\) is a factor of \(p\).
(1) \(p\) and \(p+1\) have the same number of positive factors.
(2) \(p-1\) is a factor of \(p\).
16 Sep 2014, 01:45
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Official Solution:
If \(p\) is a positive integer, is \(p\) a prime number?
(1) \(p\) and \(p+1\) have the same number of positive factors.
Primes have 2 factors, 1 and themselves, (conversely, if a positive integer has only 2 factors, it must be a prime). Thus, for the answer to be YES, both \(p\) and \(p+1\) must be primes. Are there consecutive integers that are both primes? Yes, 2 and 3.
Could there be a case where \(p\) and \(p+1\) have the same number of factors, but \(p\) is NOT a prime? Yes. For instance, both 14 (not a prime) and 15 have four factors. Similarly, both 21 (not a prime) and 22 have four factors.
Not sufficient.
(2) \(p-1\) is a factor of \(p\).
\(p-1\) and \(p\) are consecutive integers. Consecutive integers do not share any common factors other than 1. Therefore, for \(p-1\) to be a factor of \(p\), \(p-1\) must equal 1, which makes \(p\) equal to the prime number 2. Sufficient.
Answer: B
If \(p\) is a positive integer, is \(p\) a prime number?
(1) \(p\) and \(p+1\) have the same number of positive factors.
Primes have 2 factors, 1 and themselves, (conversely, if a positive integer has only 2 factors, it must be a prime). Thus, for the answer to be YES, both \(p\) and \(p+1\) must be primes. Are there consecutive integers that are both primes? Yes, 2 and 3.
Could there be a case where \(p\) and \(p+1\) have the same number of factors, but \(p\) is NOT a prime? Yes. For instance, both 14 (not a prime) and 15 have four factors. Similarly, both 21 (not a prime) and 22 have four factors.
Not sufficient.
(2) \(p-1\) is a factor of \(p\).
\(p-1\) and \(p\) are consecutive integers. Consecutive integers do not share any common factors other than 1. Therefore, for \(p-1\) to be a factor of \(p\), \(p-1\) must equal 1, which makes \(p\) equal to the prime number 2. Sufficient.
Answer: B
10 Jan 2016, 13:12
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Bunuel
What if P=1 than P-1=0 won't this make statement 2 insufficient?
