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For how many positive integers n less than or equal to 24 is n! evenly

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For how many positive integers n less than or equal to 24 is n! evenly  [#permalink]

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New post 25 Mar 2019, 00:04
00:00
A
B
C
D
E

Difficulty:

  95% (hard)

Question Stats:

13% (03:12) correct 88% (02:03) wrong based on 32 sessions

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For how many positive integers n less than or equal to 24 is n! evenly  [#permalink]

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New post 25 Mar 2019, 00:38
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Bunuel wrote:
For how many positive integers n less than or equal to 24 is n! evenly divisible by 1 + 2 + ... + n ?

(A) 8
(B) 12
(C) 16
(D) 17
(E) 21



By Property,

1 + 2 + ... + n = (1/2)*(n)(n+1)


i.e. 1*2*3*4.....(n-1)*n should be divisible by (1/2)*(n)(n+1)

i.e. n+1 should also be the factor of n! for n to be even and (n+1)/2 should be a factor of n! for n to be odd

The possible values of n may be {1, 2, 3, 4 , 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24}

Total favourable values = 16

Answer: Option C

P.S. All values preceding a prime number will be ruled out as the next integer i.e. (n+1) is prime which won't be a factor of n!
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Re: For how many positive integers n less than or equal to 24 is n! evenly  [#permalink]

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New post 07 Apr 2019, 06:58
GMATinsight wrote:
Bunuel wrote:
For how many positive integers n less than or equal to 24 is n! evenly divisible by 1 + 2 + ... + n ?

(A) 8
(B) 12
(C) 16
(D) 17
(E) 21



By Property,

1 + 2 + ... + n = (1/2)*(n)(n+1)


i.e. 1*2*3*4.....(n-1)*n should be divisible by (1/2)*(n)(n+1)

i.e. n+1 should also be the factor of n! for n to be even and (n+1)/2 should be a factor of n! for n to be odd

The possible values of n may be {1, 2, 3, 4 , 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24}

Total favourable values = 16

Answer: Option C

P.S. All values preceding a prime number will be ruled out as the next integer i.e. (n+1) is prime which won't be a factor of n!



I am not able to understand this part :


Quote:
The possible values of n may be {1, 2, 3, 4 , 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24}


if n=24, then neither (n+1) nor (n+1)/2 is a factor of 24!.

Could you please explain this a bit?

Regards,
Arup
GMAT Club Bot
Re: For how many positive integers n less than or equal to 24 is n! evenly   [#permalink] 07 Apr 2019, 06:58
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