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# If N is a positive integer, is N! divisible by 14 ?

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If N is a positive integer, is N! divisible by 14 ?  [#permalink]

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Updated on: 10 Sep 2014, 04:27
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Difficulty:

45% (medium)

Question Stats:

65% (01:46) correct 35% (01:29) wrong based on 244 sessions

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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

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Originally posted by neelesh on 09 Apr 2008, 11:44.
Last edited by Bunuel on 10 Sep 2014, 04:27, edited 2 times in total.
Renamed the topic, edited the question and added the OA.
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Re: If N is a positive integer, is N! divisible by 14 ?  [#permalink]

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12 Feb 2013, 05:49
2
7
Sachin9 wrote:
Hi Karishma/Bunuel,

Is there any other way apart from plugging in numbers?

I got this in GC CAT.
C looked suspicious so chose E.
but even if we have to plug in numbers, how can we quicky come up with numbers to say a yes and no for both statements combined?

If N is a positive integer, is N! divisible by 14 ?

In order n! to be divisible by 14=2*7, n must be at least 7. So, the question basically asks whether $$n\geq{7}$$.

(1) (N + 1)! is divisible by 15. In order (n+1)! to be divisible by 15=3*5, n+1 must be at least 5. Thus this statement implies that $$n+1\geq{5}$$ --> $$n\geq{4}$$. Not sufficient.

(2) (N + 2)! is divisible by 16. In order (n+2)! to be divisible by 16=2^4, n+2 must be at least 6 (6!=2*3*4*5*6=(2^4)*3^2*5). Thus this statement implies that $$n+2\geq{6}$$ --> $$n\geq{4}$$. Not sufficient.

(1)+(2) From above we have that $$n\geq{4}$$. If n=4, then the answer is NO but if n=7 then the answer is YES. Not sufficient.

Hope it's clear.
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Re: If N is a positive integer, is N! divisible by 14 ?  [#permalink]

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09 Apr 2008, 12:23
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
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Re: If N is a positive integer, is N! divisible by 14 ?  [#permalink]

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09 Apr 2008, 12:43
E

The least factorial where 14 is a factor is 7!, 1 x 2 x 3 x 4 x 5 x 6 x 7, so in order for 14 to be a factor of N!, we know N must be greater than 7.

1) The least value of N would be 4. (N+1)! = 5! = 1 x 2 x 3 x 4 x 5, the smallest number that is also divisible by 15. We know N is greater than or equal to 4 so 1) is a no, eliminate A and D, down to BCE

2) The least value of N would be 6. (N+2)! = 8! = 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8, the smallest number that is also divisible by 16. We know N is greater than or equal to 6 so 2) is a no, elimante B, down to C or E

Combine the two statements and we know N must be greater than 4, which could or could not be greater than 7, eliminate choice C, the answer is E
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Re: If N is a positive integer, is N! divisible by 14 ?  [#permalink]

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11 Apr 2008, 09:19
I solve this by plugging in numbers.

Ans should be 'E'

for N=4,
5! is div by 15 and 6! is div by 16 but 4! is not div by 14

for N =7,
8! is div by 15 and 9! is div by 16 and 7! is div by 14

So insuff
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Re: If N is a positive integer, is N! divisible by 14 ?  [#permalink]

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12 Feb 2013, 02:45
Hi Karishma/Bunuel,

Is there any other way apart from plugging in numbers?

I got this in GC CAT.
C looked suspicious so chose E.
but even if we have to plug in numbers, how can we quicky come up with numbers to say a yes and no for both statements combined?
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Re: If N is a positive integer, is N! divisible by 14 ?  [#permalink]

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27 Dec 2018, 04:02
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Re: If N is a positive integer, is N! divisible by 14 ?   [#permalink] 27 Dec 2018, 04:02
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