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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 factors. (2) \(p-1\) is a factor of \(p\).
(1) \(p\) and \(p+1\) have the same number of factors. (2) \(p-1\) is a factor of \(p\).
16 Sep 2014, 01:45
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Official Solution:
(1) \(p\) and \(p+1\) have the same number of factors.
Primes have 2 factors, 1 and itself, (the reverse is also true: if a positive integer has 2 factors, then it must be a prime). So, for the answer to the question to be YES, both \(p\) and \(p+1\) must be primes. Are there consecutive primes? Yes, 2 and 3.
Could we have a case when \(p\) and \(p+1\) have the same number of factors, and \(p\) is NOT a prime? Yes. For example, both 14 (not a prime) and 15 have four factors. Also, 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 factor but 1. Therefore, \(p-1\) to be a factor of \(p\), \(p-1\) must be 1, which makes \(p\) equal to prime number 2. Sufficient.
Answer: B
(1) \(p\) and \(p+1\) have the same number of factors.
Primes have 2 factors, 1 and itself, (the reverse is also true: if a positive integer has 2 factors, then it must be a prime). So, for the answer to the question to be YES, both \(p\) and \(p+1\) must be primes. Are there consecutive primes? Yes, 2 and 3.
Could we have a case when \(p\) and \(p+1\) have the same number of factors, and \(p\) is NOT a prime? Yes. For example, both 14 (not a prime) and 15 have four factors. Also, 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 factor but 1. Therefore, \(p-1\) to be a factor of \(p\), \(p-1\) must be 1, which makes \(p\) equal to prime number 2. Sufficient.
Answer: B
25 Sep 2014, 05:14
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Official Solution:
(1) p and p+1 have the same number of factors.
Primes have 2 factors, 1 and itself, (the reverse is also true: if a positive integer has 2 factors, then it must be a prime). So, for the answer to the question to be YES, both p and p+1 must be primes. Are there consecutive primes? Yes, 2 and 3.
Could we have a case when p and p+1 have the same number of factors, and p is NOT a prime? Yes. For example, both 14 (not a prime) and 15 have four factors. Also, 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 factor but 1. Therefore, p-1 to be a factor of p, p-1 must be 1, which makes p equal to prime number 2. Sufficient.
Answer: B
Bunuel
BUt for condition 2.
Any prime no greater than 2 cannot have p-1 as a factor.
So dont you think that condn 2 gives both YES?NO Answer-------- hence not sufficient.
Please let me know what I m missing.
Thanks
(1) p and p+1 have the same number of factors.
Primes have 2 factors, 1 and itself, (the reverse is also true: if a positive integer has 2 factors, then it must be a prime). So, for the answer to the question to be YES, both p and p+1 must be primes. Are there consecutive primes? Yes, 2 and 3.
Could we have a case when p and p+1 have the same number of factors, and p is NOT a prime? Yes. For example, both 14 (not a prime) and 15 have four factors. Also, 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 factor but 1. Therefore, p-1 to be a factor of p, p-1 must be 1, which makes p equal to prime number 2. Sufficient.
Answer: B
Bunuel
BUt for condition 2.
Any prime no greater than 2 cannot have p-1 as a factor.
So dont you think that condn 2 gives both YES?NO Answer-------- hence not sufficient.
Please let me know what I m missing.
Thanks
