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16 Sep 2014, 01:17
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If \(n\) is a positive integer, is \(n!\) divisible by 14 ?
(1) \((n + 1)!\) is divisible by 15.
(2) \((n + 2)!\) is divisible by 16.
(1) \((n + 1)!\) is divisible by 15.
(2) \((n + 2)!\) is divisible by 16.
16 Sep 2014, 01:17
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Official Solution:
If \(n\) is a positive integer, is \(n!\) divisible by 14 ?
In order for \(n!\) to be divisible by 14 (which is 2 * 7), \(n\) must be at least 7. So, the question is essentially asking whether \(n \geq 7\).
(1) \((n + 1)!\) is divisible by 15.
In order for \((n + 1)!\) to be divisible by 15 (which is 3 * 5), \(n + 1\) must be at least 5. Thus, this statement implies that \(n + 1 \geq 5\) and, therefore, \(n \geq 4\). This is not sufficient.
(2) \((n + 2)!\) is divisible by 16.
In order for \((n + 2)!\) to be divisible by 16 (which is \(2^4\)), \(n + 2\) must be at least 6 (since \(6! = 2 * 3 * 4 * 5 * 6 = (2^4)*(3^2)*5\)). Thus, this statement implies that \(n + 2 \geq 6\) and, therefore, \(n \geq 4\). This is not sufficient.
(1) + (2) From the above, we know that \(n \geq 4\). If \(n = 4\), then the answer is NO; but if \(n = 7\), then the answer is YES. Not sufficient.
Answer: E
If \(n\) is a positive integer, is \(n!\) divisible by 14 ?
In order for \(n!\) to be divisible by 14 (which is 2 * 7), \(n\) must be at least 7. So, the question is essentially asking whether \(n \geq 7\).
(1) \((n + 1)!\) is divisible by 15.
In order for \((n + 1)!\) to be divisible by 15 (which is 3 * 5), \(n + 1\) must be at least 5. Thus, this statement implies that \(n + 1 \geq 5\) and, therefore, \(n \geq 4\). This is not sufficient.
(2) \((n + 2)!\) is divisible by 16.
In order for \((n + 2)!\) to be divisible by 16 (which is \(2^4\)), \(n + 2\) must be at least 6 (since \(6! = 2 * 3 * 4 * 5 * 6 = (2^4)*(3^2)*5\)). Thus, this statement implies that \(n + 2 \geq 6\) and, therefore, \(n \geq 4\). This is not sufficient.
(1) + (2) From the above, we know that \(n \geq 4\). If \(n = 4\), then the answer is NO; but if \(n = 7\), then the answer is YES. Not sufficient.
Answer: E
09 May 2023, 04:22
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I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
