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If 5^a is a factor of n!, and the greatest integer value of a is 6, wh Updated on: 14 Sep 2018, 09:50
Originally posted by patnaiksonal on 14 Sep 2018, 09:15.
Last edited by patnaiksonal on 14 Sep 2018, 09:50, edited 1 time in total.
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54% (01:53) correct 46% (02:04) wrong based on 412 sessions

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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
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C
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souvonik2k
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If 5^a is a factor of n!, and the greatest integer value of a is 6, wh 14 Sep 2018, 09:56
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patnaiksonal
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
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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 Updated on: 19 Sep 2018, 09:15
Originally posted by arjunnath on 14 Sep 2018, 22:18.
Last edited by arjunnath on 19 Sep 2018, 09:15, edited 1 time in total.
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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
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If 5^a is a factor of n!, and the greatest integer value of a is 6, wh 19 Sep 2018, 09:02
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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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CAMANISHPARMAR
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If 5^a is a factor of n!, and the greatest integer value of a is 6, wh 19 Sep 2018, 09:05
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Natty97
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
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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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If 5^a is a factor of n!, and the greatest integer value of a is 6, wh 19 Sep 2018, 09:38
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patnaiksonal
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
Show more

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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If 5^a is a factor of n!, and the greatest integer value of a is 6, wh 20 Sep 2018, 09:34
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Natty97
Thanks for pointing out my mistake CAMANISHPARMAR
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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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If 5^a is a factor of n!, and the greatest integer value of a is 6, wh 08 Nov 2020, 09:10
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patnaiksonal
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!?
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A mathematics question can never be written this way. I don't know the source, but I would not recommend using it for study. The question is treating n as if it's some kind of variable, but in a math question written this way, n represents some unknown specific number (and the same is true for the letter a in this question). As written, the question cannot be answered - the answer might be 3 or 4.

It's easy to see what's wrong with this question (and this also illustrates why this issue is important for GMAT purposes) if you rephrase it as a DS question:

If b and n are positive integers, what is the value of b?
1. 5^6 is a factor of n!, but 5^7 is not a factor of n!
2. b is the largest integer for which 7^b is a factor of n!


Statement 1 ensures n is between 25 and 29 inclusive, and Statement 2 tells us we're looking for the largest power of 7 that will divide n!. But the value of b is 3 when n = 25, 26 or 27, and is 4 when n = 28 or 29, so we can't answer the question. There is no logical reason, looking at Statement 2, to conclude that n takes on its maximum possible value (though if you look back at the PS question, that's precisely the illogical conclusion we're supposed to draw). So the answer is E.

If this DS question can't be answered, the PS question in this thread can't be answered either.
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If 5^a is a factor of n!, and the greatest integer value of a is 6, wh 21 Nov 2021, 00:43
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patnaiksonal
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
Show more

Hi, Can someone explain this question in detail not able to infer. Have gone through the explanation by others. However the answer is not convincing.

Thanks
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If 5^a is a factor of n!, and the greatest integer value of a is 6, wh 10 Apr 2025, 09:56
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5^a is a factor of n!, and the greatest integer value of a is 6. Let's count the 5 in a factorial till be get 6 5s

1,2,3,4,5...,10,..15,...,20,...25
---------1----2----3-----4-----6 (25 has 2 5s)

So the max value of n could be 29, as with 30, we will get another 5

For largest value of b, let's take n = 29
29/7 = 4

Answer 4
patnaiksonal
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
Show more
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