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If n is a positive integer and n! is a multiple of 7^6 but not 7^9, th

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Bunuel
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If n is a positive integer and n! is a multiple of 7^6 but not 7^9, th [#permalink] If n is a positive integer and n! is a multiple of 7^6 but not 7^9, th   14 Apr 2020, 08:48
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If n is a positive integer and \(n!\) is a multiple of \(7^6\) but not \(7^9\), then how many value can n take ?

A. 21
B. 14
C. 13
D. 12
E. 7


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If n is a positive integer and n! is a multiple of 7^6 but not 7^9, th [#permalink] If n is a positive integer and n! is a multiple of 7^6 but not 7^9, th   Updated on: 14 Apr 2020, 10:41
Bunuel
If n is a positive integer and \(n!\) is a multiple of \(7^6\) but not \(7^9\), then how many value can n take ?

A. 21
B. 14
C. 13
D. 12
E. 7


when n! is a multiple of \(7^6\), minimum value of n is 42.
when n! is a multiple of \(7^9\), minimum value of n is 56.
number of values of n = 56 - 42 = 14
Ans: B

Originally posted by ArunSharma12 on 14 Apr 2020, 10:13.
Last edited by ArunSharma12 on 14 Apr 2020, 10:41, edited 1 time in total.
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Re: If n is a positive integer and n! is a multiple of 7^6 but not 7^9, th [#permalink] If n is a positive integer and n! is a multiple of 7^6 but not 7^9, th   14 Apr 2020, 10:35
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Bunuel
If n is a positive integer and \(n!\) is a multiple of \(7^6\) but not \(7^9\), then how many value can n take ?

A. 21
B. 14
C. 13
D. 12
E. 7


Are You Up For the Challenge: 700 Level Questions

Asked: If n is a positive integer and \(n!\) is a multiple of \(7^6\) but not \(7^9\), then how many value can n take ?

Maximum power of 7 in 56! = 8 + 1 = 9
Maximum power of 7 in 42! = 6
41<n<56

n = {42,43,44,45,46,47,48,49,50,51,52,53,54,55} : 14 values on n are feasible

IMO B
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If n is a positive integer and n! is a multiple of 7^6 but not 7^9, th [#permalink] If n is a positive integer and n! is a multiple of 7^6 but not 7^9, th   23 Apr 2020, 21:56
Each multiple of 7 will add one 7 to the factorial, but 49 will add two 7 to the factorial.

i.e. 7.1, 7.2, 7.3, 7.4, 7.5, 7.6, 7.7, 7.8

Now n! has to have 6 powers of 7, but not 9. And 55! will have same number of 7 as 49!.

So n can take values from 42 to 55, both inclusive.

55-42+1 = 14
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Re: If n is a positive integer and n! is a multiple of 7^6 but not 7^9, th [#permalink] If n is a positive integer and n! is a multiple of 7^6 but not 7^9, th   21 Sep 2024, 06:24
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Re: If n is a positive integer and n! is a multiple of 7^6 but not 7^9, th [#permalink]
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