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

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

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.

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 \(n!= \frac{29}{7}=4\)

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!

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.

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
Ritesh

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