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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
M22-32
(1) (N + 1)! is divisible by 15
(2) (N + 2)! is divisible by 16
M22-32
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12 Feb 2013, 05:49
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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
Hope it's clear.
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
Hope it's clear.
General Discussion
09 Apr 2008, 12:23
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looks to be E..
(N+2)! =16N basically means we have 2^4 of 2s at a minimum..that means N+2=6 at the least..or N at the least equal 4.. again we cant say anything about N
(N+2)! =16N basically means we have 2^4 of 2s at a minimum..that means N+2=6 at the least..or N at the least equal 4.. again we cant say anything about N
