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# If N is a positive integer, is N! divisible by 14? 1) (N+1)!

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If N is a positive integer, is N! divisible by 14? 1) (N+1)! [#permalink]

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13 Dec 2006, 21:52
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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

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13 Dec 2006, 22:05
E

n! can be divided by 14 (or 7 x2)
n must be at least 7 in order to satisfy that condition
because 7! = 7.6.5.4.3.2.1 is the least factorial that can be divided by 7x2
and those lower cannot.

I) (n+1)! can be divided by 15 (or 5x3)
This means n+1 must be at least 5.
which means n>=4

Not sufficient

II) (n+2)! can be divided by 16 ( or 2x2x2x2)
This means (n+2)! must be at least 6!=1.2.3.4.5.6
which means n>=4

Not sufficient. Same conclusion as I).

I) and II) together tells us that n! must be at least 6! because
n!=6.5.4.3.2.1 is the least integer that can be divided both by 15 and 14.

Still not sufficient...
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Last edited by Viperace on 13 Dec 2006, 22:10, edited 1 time in total.

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13 Dec 2006, 22:14
Viperace wrote:
E

n! can be divided by 14 (or 7 x2)
n must be at least 7 in order to satisfy that condition
because 7! = 7.6.5.4.3.2.1 is the least factorial that can be divided by 7x2
and those lower cannot.

I) (n+1)! can be divided by 15 (or 5x3)
This means n+1 must be at least 5.
which means n>=4

Not sufficient

II) (n+2)! can be divided by 16 ( or 2x2x2x2)
This means (n+2)! must be at least 6!=1.2.3.4.5.6
which means n>=4

Not sufficient. Same conclusion as I).

I) and II) together tells us that n! must be at least 6! because
n!=6.5.4.3.2.1 is the least integer that can be divided both by 15 and 14.

Still not sufficient...

I was about to post D when I saw viperace's post.

I made the dumb mistake of assuming that (N+1)! is 15!
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13 Dec 2006, 22:14
E too...

Same reasoning

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13 Dec 2006, 22:14
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# If N is a positive integer, is N! divisible by 14? 1) (N+1)!

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