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03 Sep 2014, 08:44
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A
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Difficulty:
25%
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Question Stats:
69% (00:58) correct
31%
(01:01)
wrong
based on 270
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If \(n\) is a positive integer, is \(n!\) divisible by 66?
(1) \(n!\) is divisible by 11
(2) \(n!\) is divisible by 12
(1) \(n!\) is divisible by 11
(2) \(n!\) is divisible by 12
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Pure algebraic questions like this one are no longer part of the GMAT Data Sufficiency syllabus.
For relevant Data Sufficiency questions, please check the Data Sufficiency forum:
Data Sufficiency Forum
Data Sufficiency Forum
03 Sep 2014, 09:10
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goodyear2013
\(66 = 2 * 3 * 11\)
So, \(N!\) is divisible by 66 if and only if it has 2, 3 & 11 as factors.
Furthermore, if \(N!\) is divisible by a prime number, it has to be divisible by every number smaller than the prime. This is true because we can't multiply any two numbers to get the prime.
Let's say \(X\) is a prime factor of \(N!\). This means \(N! = N * (N - 1) * (N - 2) *....* X * (X - 1) *....* 3 * 2 * 1\)
1) \(N!\) is divisible by \(11\). So, \(N!\) is divisible by \(6\) as well. ---> Sufficient
2) \(N!\) is divisible by \(12\). Now, it might be that \(N!\) has \(12\) as a factor. Or it might just have \(3\) & \(4\) as factors. (\(4! = 24\) is divisible by \(12\) as well. and so are \(5!, 6! & 7!\)) ---> Not Sufficient
Hence, the answer is A.
Hope that makes it clear.
12 Oct 2023, 00:39
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If \(n\) is a positive integer, is \(n!\) divisible by 66?
(1) \(n!\) is divisible by 11
Since 11 is prime, for \(n!\) to be divisible by 11, \(n\) must be 11 or greater. The product \(1*2*3*...*n\) is divisible by both 11 and 6, making it divisible by 66. Sufficient.
(2) \(n!\) is divisible by 12
If \(n = 12\), then \(n!\) is divisible by 66. However, if \(n = 4\), then \(n!\) is not divisible by 66. Not sufficient.
Answer: A
(1) \(n!\) is divisible by 11
Since 11 is prime, for \(n!\) to be divisible by 11, \(n\) must be 11 or greater. The product \(1*2*3*...*n\) is divisible by both 11 and 6, making it divisible by 66. Sufficient.
(2) \(n!\) is divisible by 12
If \(n = 12\), then \(n!\) is divisible by 66. However, if \(n = 4\), then \(n!\) is not divisible by 66. Not sufficient.
Answer: A
