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12 Sep 2022, 11:26
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C
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Difficulty:
65%
(hard)
Question Stats:
46% (01:40) correct
54%
(01:57)
wrong
based on 90
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If \(p\) is a positive integer, what is the value of \(p\) ?
(1) \(p\) does not have a factor \(k\) such that \(1 < k < p\).
(2) \(p((p - 1)! + 1)\) is divisible by \(2p\).
(1) \(p\) does not have a factor \(k\) such that \(1 < k < p\).
(2) \(p((p - 1)! + 1)\) is divisible by \(2p\).
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12 Sep 2022, 12:00
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Point 1 is true for all prime numbers.
For Point 2 to be true, the expression has to be divisible by 2P Or ((P-1)! +1) has to be divisible by 2, I.e. an even number. Hence, (P-1)! need to be odd number. For (P-1) >=2 the factorial is an even number. Hence, P must be less than 3. Being a positive integer it can be either 1 or 2.
Hence, both the conditions are not independently suffecient. But taken together we can have 2, the positive integer and a prime number fulfilling both the conditions.
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For Point 2 to be true, the expression has to be divisible by 2P Or ((P-1)! +1) has to be divisible by 2, I.e. an even number. Hence, (P-1)! need to be odd number. For (P-1) >=2 the factorial is an even number. Hence, P must be less than 3. Being a positive integer it can be either 1 or 2.
Hence, both the conditions are not independently suffecient. But taken together we can have 2, the positive integer and a prime number fulfilling both the conditions.
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12 Sep 2022, 12:24
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Statement 1
The first statement indicates that p is a prime number.
However it can be any prime number, hence the statement is not sufficient.
Statement 2
From the second statement we can have p = 2 or p = 1.
Hence that's not sufficient as well.
Combining
Once we combine the two statements, the only value of p that's satisfies is p = 2
IMO C
The first statement indicates that p is a prime number.
However it can be any prime number, hence the statement is not sufficient.
Statement 2
From the second statement we can have p = 2 or p = 1.
Hence that's not sufficient as well.
Combining
Once we combine the two statements, the only value of p that's satisfies is p = 2
IMO C
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