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#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # If 100!/x is not an integer, which of the following could be the value

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Math Expert V
Joined: 02 Sep 2009
Posts: 56303
If 100!/x is not an integer, which of the following could be the value  [#permalink]

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18 00:00

Difficulty:   45% (medium)

Question Stats: 69% (01:39) correct 31% (01:39) wrong based on 287 sessions

### HideShow timer Statistics If 100!/x is not an integer, which of the following could be the value of x?

A. 5^24
B. 7^16
C. 11^9
D. 13^6
E. 17^6

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SVP  B
Joined: 06 Nov 2014
Posts: 1877
Re: If 100!/x is not an integer, which of the following could be the value  [#permalink]

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5
2
Bunuel wrote:
If 100!/x is not an integer, which of the following could be the value of x?

A. 5^24
B. 7^16
C. 11^9
D. 13^6
E. 17^6

100!/x is not an integer, this means that x has certain factors that are not common with 100!
To find out the total powers of a number in a factorial, follow this:

[100/5] +[100/5^2] + [100/5^3] = 20 + 4 + 0, where [] = integral part.
Similarly for 7, [100/7] + [100/7^2] + [100/7^3] = 14 + 2 + 0 = 16
For 11, [100/11] + [100/11^2] = 9 + 0
For 13, [100/13] + [100/13^2] = 7 + 0
For 17, [100/17] + [100/17^2] = 5 + 0

Hence the maximum power of 17 in 100! is 5
Therefore 100!/17^6 will not be an integer.

Correct Option: E
##### General Discussion
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Re: If 100!/x is not an integer, which of the following could be the value  [#permalink]

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100! has 24 5's, 16 7's, 9 11's, 7 13's and 5 17's.

100! is not completely divisible by 17^6 as 100! has only 5 17's.

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If 100!/x is not an integer, which of the following could be the value  [#permalink]

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Bunuel wrote:
If 100!/x is not an integer,which of the following could be the value of x?

A. 5^24
B. 7^16
C. 11^9
D. 13^6
E. 17^6

Checking the options is the best way to go. The answer will be a number which doesn't completely divide 100!

No of 5's in 100! = 24
No of 7's in 100! = 16
No of 11's in 100! = 9
No of 13's in 100! = 7

No of 17's in 100! = 5

There are 5 seventeens in 100! , however we are dividing by 17^6 , so this can not divide 100!

Hence our answer will be E

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Senior Manager  P
Joined: 27 Dec 2016
Posts: 332
Re: If 100!/x is not an integer, which of the following could be the value  [#permalink]

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Hi,

Can anyone please help me in regards to finding out 5s, 7s, 11s, 13s , and 17s in 100!. How exactly does one know that 100! has 5 24s, not 25 or has 5 17s not 6?

Thank You!
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Re: If 100!/x is not an integer, which of the following could be the value  [#permalink]

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1
Bunuel wrote:
If 100!/x is not an integer, which of the following could be the value of x?

A. 5^24
B. 7^16
C. 11^9
D. 13^6
E. 17^6

17, 34, 51, 68, and 85. Thus, we have 5 multiples of 17 in 100!.

Therefore, 100!/(17^6) IS NOT an integer.

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Manager  G
Joined: 01 Feb 2017
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If 100!/x is not an integer, which of the following could be the value  [#permalink]

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If 100!/x is a non-interger, then x must be larger than 100!
Let's look at all the choices to find an option with value greater than 100!, Using concept: power of prime number in n!= n/p + n/p^2 + .......n/p^k, where p^k =<n.

A) Power of 5: 100/5 + 100/5^2 = 20 + 4 = 24. Therfore, 5^24. Same as the option. Hence, incorrect.

B) Power of 7: 100/7 + 100/7^2 = 14 + 2 = 16. Therfore, 7^16. Same as the option. Hence, incorrect.

C) Power of 11: 100/11= 9. Therfore, 11^9. Same as the option. Hence, incorrect.

D) Power of 13: 100/13 = 7. Therfore, 13^7. Smaller than the option. Hence, incorrect.

E) Power of 17: 100/17 = 5. Therfore, 17^5. Larger than the option. Hence, CORRECT
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Re: If 100!/x is not an integer, which of the following could be the value  [#permalink]

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_________________ Re: If 100!/x is not an integer, which of the following could be the value   [#permalink] 24 Aug 2018, 11:20
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