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For a positive integer n, if 5^n is a factor of 25!, but 5^n+1

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Math Revolution GMAT Instructor
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For a positive integer n, if 5^n is a factor of 25!, but 5^n+1 [#permalink]

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19 Sep 2017, 00:09
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[GMAT math practice question]

For a positive integer n, if $$5^n$$ is a factor of 25!, but $$5^{n+1}$$ is not a factor of 25!, what is the value of n?

A. 5
B. 6
C. 7
D. 8
E. 9
[Reveal] Spoiler: OA

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Re: For a positive integer n, if 5^n is a factor of 25!, but 5^n+1 [#permalink]

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19 Sep 2017, 00:26
MathRevolution wrote:
[GMAT math practice question]

For a positive integer n, if $$5^n$$ is a factor of 25!, but $$5^{n+1}$$ is not a factor of 25!, what is the value of n?

A. 5
B. 6
C. 7
D. 8
E. 9

Similar question also posted by you: https://gmatclub.com/forum/for-a-positi ... 27957.html
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Re: For a positive integer n, if 5^n is a factor of 25!, but 5^n+1 [#permalink]

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19 Sep 2017, 03:15
MathRevolution wrote:
[GMAT math practice question]

For a positive integer n, if $$5^n$$ is a factor of 25!, but $$5^{n+1}$$ is not a factor of 25!, what is the value of n?

A. 5
B. 6
C. 7
D. 8
E. 9

25! has five 5s.
because 25/5 = 5
hence 5^5 will completely divide 25! but not 5^6
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Re: For a positive integer n, if 5^n is a factor of 25!, but 5^n+1 [#permalink]

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19 Sep 2017, 12:36
MathRevolution wrote:
[GMAT math practice question]

For a positive integer n, if $$5^n$$ is a factor of 25!, but $$5^{n+1}$$ is not a factor of 25!, what is the value of n?

A. 5
B. 6
C. 7
D. 8
E. 9

This question basically asks us to find the find what could be the maximum value of n OR one can say that what is the highest multiple of 5 which can divide 25! completely.

There are two ways to find the number of 5s in 25! to solve this -

1. Check factors of 5s in 25! :

5,10,15,20,25

The total number of 5s (n) are = 1 + 1+ 1 + 1 + 2 = 6

Thus n = 6, because n +1 = 7 and $$5^n+1$$ would not be a factor of 25!.

2. Division method of finding the number of 5s.

25/5 = 5

5/5 = 1

Thus total 5s are = 5 + 1 = 6.

The correct answer is Option B.
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Re: For a positive integer n, if 5^n is a factor of 25!, but 5^n+1 [#permalink]

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19 Sep 2017, 12:42
souvonik2k wrote:
MathRevolution wrote:
[GMAT math practice question]

For a positive integer n, if $$5^n$$ is a factor of 25!, but $$5^{n+1}$$ is not a factor of 25!, what is the value of n?

A. 5
B. 6
C. 7
D. 8
E. 9

25! has five 5s.
because 25/5 = 5
hence 5^5 will completely divide 25! but not 5^6

A slight mistake I guess.

25/5 = 5, but then 5/5 = 1 also needs to be done.

So total 5s in 25! would be 6.

Had the number be say 15! or 18! then we could have done a single division and that would have been suffice to get the number of 5s.

Regards,
Squib
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Re: For a positive integer n, if 5^n is a factor of 25!, but 5^n+1 [#permalink]

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21 Sep 2017, 00:58
=>

25! = 1 x 2 x … x 5 x … x 10 x … x 15 x … x 20 x … x 25
= 1 x … x 4 x 6 x … x 9 x 11 x … x 14 x 16 x … x 19 x 21 x … x 24 x 5 x 10 x 15 x 20 x 25
= 1 x … x 4 x 6 x … x 9 x 11 x … x 14 x 16 x … x 19 x 21 x … x 24 x ( 5 x 5 x 2 x 5 x 3 x 5 x 4 x 5^2 )
= 1 x … x 4 x 6 x … x 9 x 11 x … x 14 x 16 x … x 19 x 21 x … x 24 x (2 x 3 x 4 x 5^6 )

Thus n = 6.

Ans: B
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Re: For a positive integer n, if 5^n is a factor of 25!, but 5^n+1   [#permalink] 21 Sep 2017, 00:58
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