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16 Sep 2014, 00:12
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If \(a\), \(b\), and \(c\) are integers and \(a < b < c\), 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, 00:12
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Official Solution:
If \(a\), \(b\), and \(c\) are integers and \(a < b < c\), are \(a\), \(b\), and \(c\) consecutive integers?
Note that:
1. The factorial of a negative number is undefined.
2. \(0!=1\).
3. Only two factorials are odd: \(0!=1\) and \(1!=1\).
4. If \(x!\) is a prime number, then \(x = 2\) is the only solution.
(1) The median of \((a!, \ b!, \ c!)\) is an odd number.
This implies that \(b!\) is odd. Therefore, \(b\) must be either 0 or 1. However, if \(b = 0\), then \(a\) would be a negative number (since \(a < b\)), making \(a!\) undefined, which is not possible. Therefore, \(a = 0\) and \(b = 1\) (yielding \((a!, \ b!, \ c!)=(1, 1, c!)\)). If \(c = 2\), the answer is YES, but if \(c\) is any other number, the answer is NO. Not sufficient.
(2) \(c!\) is a prime number.
This implies that \(c=2\). Not sufficient.
(1) + (2) From the previous analysis, we know that \(a = 0\), \(b = 1\), and \(c = 2\). Thus, the answer to the question is YES. Sufficient.
Answer: C
If \(a\), \(b\), and \(c\) are integers and \(a < b < c\), are \(a\), \(b\), and \(c\) consecutive integers?
Note that:
1. The factorial of a negative number is undefined.
2. \(0!=1\).
3. Only two factorials are odd: \(0!=1\) and \(1!=1\).
4. If \(x!\) is a prime number, then \(x = 2\) is the only solution.
(1) The median of \((a!, \ b!, \ c!)\) is an odd number.
This implies that \(b!\) is odd. Therefore, \(b\) must be either 0 or 1. However, if \(b = 0\), then \(a\) would be a negative number (since \(a < b\)), making \(a!\) undefined, which is not possible. Therefore, \(a = 0\) and \(b = 1\) (yielding \((a!, \ b!, \ c!)=(1, 1, c!)\)). If \(c = 2\), the answer is YES, but if \(c\) is any other number, the answer is NO. Not sufficient.
(2) \(c!\) is a prime number.
This implies that \(c=2\). Not sufficient.
(1) + (2) From the previous analysis, we know that \(a = 0\), \(b = 1\), and \(c = 2\). Thus, the answer to the question is YES. Sufficient.
Answer: C
11 Jan 2015, 11:18
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GMAT888
That's NOT correct. For (2) we don't know whether a and b are positive integers. WE cannot use the info from (1) when considering the second statement.
