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If 5^a is a factor of n!, and the greatest integer value of a is 6, wh

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If 5^a is a factor of n!, and the greatest integer value of a is 6, wh  [#permalink]

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New post Updated on: 14 Sep 2018, 08:50
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If 5^a is a factor of n!, and the greatest integer value of a is 6, what is the largest possible value of b so that 7^b is a factor of n!?

(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

Originally posted by patnaiksonal on 14 Sep 2018, 08:15.
Last edited by patnaiksonal on 14 Sep 2018, 08:50, edited 1 time in total.
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If 5^a is a factor of n!, and the greatest integer value of a is 6, wh  [#permalink]

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New post 14 Sep 2018, 08:56
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patnaiksonal wrote:
If 5^a is a factor of n!, and the greatest integer value of a is 6, what is the largest possible value of b so that 7^b is a factor of n!?

(A) 2
(B) 3
(C) 4
(D) 5
(E) 6


Assuming a value for n, suppose n = 25

Number of 5s in 25! = \(\frac{25}{5}\) + \(\frac{25}{{5^2}}\) = 5+1 = 6

The maximum value of n can be 29 because if n=30, no. of 5s become \(\frac{30}{5}\) + \(\frac{30}{{5^2}}\) = 6+1 = 7

The largest value of b so that 7^b is a factor of 29! is \(\frac{29}{7}\) = 4

Answer C.
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If 5^a is a factor of n!, and the greatest integer value of a is 6, wh  [#permalink]

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New post Updated on: 19 Sep 2018, 08:15
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If 5^6 is a factor of n i.e. n! has to have 5*5*5*5*5*5 in it.
as a result 25=<n=<29 to include every 5 (of 5^6) in it.
In the form of: 5,10,15,20,25

And we have take 29 as it can not be 30 since that would mean 5^7.

Now once we know in where n could lie, we see the second part of the question, which asks us the value of b in 7^b
That is the greatest possible value for 7^b in 25=<n=<29
Now lets check, all values of 7 possible with n are: 7,14,21,28 i.e 4 values of 7

Therefore the largest possible value of b is 4

Answer is C which is 4


Thanks for pointing out my mistake CAMANISHPARMAR

Originally posted by Natty97 on 14 Sep 2018, 21:18.
Last edited by Natty97 on 19 Sep 2018, 08:15, edited 1 time in total.
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Re: If 5^a is a factor of n!, and the greatest integer value of a is 6, wh  [#permalink]

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New post 19 Sep 2018, 08:02
Given the constraints since the greatest integer value of a is 6 - n could take max value of 29.

For 29! to be divisivle by 7^b - the max value which b can take is 4.

Correct Ans is option C
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Re: If 5^a is a factor of n!, and the greatest integer value of a is 6, wh  [#permalink]

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New post 19 Sep 2018, 08:05
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Natty97 wrote:
It is very simple. Just try and understand the concept.
If 5^6 is a factor of n i.e. n! has to have 5*5*5*5*5*5 in it.
as a result 30=<n=<34 to include every 5 (of 5^6) in it.
In the form of: 5,10,15,20,25,30


Don't forget 25 is 5^2. Hence the max value of n could be 29. Think over it, in case if you have further doubts, please feel free to revert.
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Re: If 5^a is a factor of n!, and the greatest integer value of a is 6, wh  [#permalink]

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New post 19 Sep 2018, 08:38
patnaiksonal wrote:
If 5^a is a factor of n!, and the greatest integer value of a is 6, what is the largest possible value of b so that 7^b is a factor of n!?

(A) 2
(B) 3
(C) 4
(D) 5
(E) 6


OA:C

5! = 5*4*3*2*1 (\(5!\) is divisible by \(5^1\))
10! =10*9*.....*5*4*3*2*1 (\(10!\) is divisible by \(5^2\))
15! =15*14*13*........*10*9*.....*5*4*3*2*1 (\(15!\) is divisible by \(5^3\))
20! =20*19*....*15*14*13*........*10*9*.....*5*4*3*2*1 (\(20!\) is divisible by \(5^4\))
25! =25*24*....*20*19*....*15*14*13*........*10*9*.....*5*4*3*2*1 (\(25!\) is divisible by \(5^6\))
30!=30*29*...*25*24*....*20*19*....*15*14*13*........*10*9*.....*5*4*3*2*1 (\(30!\) is divisible by \(5^7\))

largest value of \(n\) such \(5^6\) is a factor of \(n!\) is \(29\).

largest possible value of \(b\) so that \(7^b\) is a factor of \(n!= \frac{29}{7}=4\)
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Re: If 5^a is a factor of n!, and the greatest integer value of a is 6, wh  [#permalink]

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New post 20 Sep 2018, 08:34
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Natty97 wrote:
Thanks for pointing out my mistake CAMANISHPARMAR


You are most welcome. Please feel free to get in touch with me in case if I could be of any other help! Wishing you all the very best for your GMAT!
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Re: If 5^a is a factor of n!, and the greatest integer value of a is 6, wh &nbs [#permalink] 20 Sep 2018, 08:34
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