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16 Sep 2014, 01:44
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If \(a\), \(b\), and \(c\) are integers and \(a \lt b \lt c\), are \(a\), \(b\), and \(c\) consecutive integers?
(1) The median of \(\{a!, \ b!, \ c!\}\) is an odd number.
(2) \(c!\) is a prime number.
(1) The median of \(\{a!, \ b!, \ c!\}\) is an odd number.
(2) \(c!\) is a prime number.
16 Sep 2014, 01:44
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Official Solution:
Note that:
A. The factorial of a negative number is undefined.
B. \(0!=1\).
C. Only two factorials are odd: \(0!=1\) and \(1!=1\).
D. Factorial of a number which is prime is \(2!=2\).
(1) The median of \(\{a!, \ b!, \ c!\}\) is an odd number. This implies that \(b!=odd\). Thus \(b\) is 0 or 1. But if \(b=0\), then \(a\) is a negative number, so in this case \(a!\) is not defined. Therefore \(a=0\) and \(b=1\), so the set is \(\{0!, \ 1!, \ c!\}=\{1, \ 1, \ c!\}\). Now, if \(c=2\), then the answer is YES but if \(c\) is any other number then the answer is NO. Not sufficient.
(2) \(c!\) is a prime number. This implies that \(c=2\). Not sufficient.
(1)+(2) From above we have that \(a=0\), \(b=1\) and \(c=2\), thus the answer to the question is YES. Sufficient.
Answer: C
Note that:
A. The factorial of a negative number is undefined.
B. \(0!=1\).
C. Only two factorials are odd: \(0!=1\) and \(1!=1\).
D. Factorial of a number which is prime is \(2!=2\).
(1) The median of \(\{a!, \ b!, \ c!\}\) is an odd number. This implies that \(b!=odd\). Thus \(b\) is 0 or 1. But if \(b=0\), then \(a\) is a negative number, so in this case \(a!\) is not defined. Therefore \(a=0\) and \(b=1\), so the set is \(\{0!, \ 1!, \ c!\}=\{1, \ 1, \ c!\}\). Now, if \(c=2\), then the answer is YES but if \(c\) is any other number then the answer is NO. Not sufficient.
(2) \(c!\) is a prime number. This implies that \(c=2\). Not sufficient.
(1)+(2) From above we have that \(a=0\), \(b=1\) and \(c=2\), thus the answer to the question is YES. Sufficient.
Answer: C
12 Jan 2015, 05:10
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Bunuel
Query w.r.t Stmnt 1
I understand the highlighted part - where a! is not defined. But can we officially consider such a scenario where mathematically it is not possible to derive it?
Is this the reason that in Stmnt 1 , we only consider the Red part? And the actual answer (Yes/No) depends on the vaue of "C" ?
