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If 17!/7^m is an integer, what is the greatest possible value of m ?

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Bunuel
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If 17!/7^m is an integer, what is the greatest possible value of m ? [#permalink] New post  23 Sep 2022, 06:14
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If 17!/7^m is an integer, what is the greatest possible value of m ?

A. 2
B. 3
C. 4
D. 5
E. 6
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Anton1234
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If 17!/7^m is an integer, what is the greatest possible value of m ? [#permalink] New post  01 Oct 2022, 06:43
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The way I did this was finding the prime factorisation of 17!

1*2*3*4*5*6*7...*14

then solve for prime factorisation so it gives you 7^2

as 14/2 = 7 then 7 is the only prime left

so you have 2 7's in prime factorisation one from the number 7 and one from 14

Doing this gets you 7^2

Therefore 17! is divisible by 7^2

I don't know if this is the correct way of doing it, worked for me however
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nicbo
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Re: If 17!/7^m is an integer, what is the greatest possible value of m ? [#permalink] New post  29 Mar 2023, 03:09
Bunuel KarishmaB could you please explain this approach? Is there another approach that I can take?
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Re: If 17!/7^m is an integer, what is the greatest possible value of m ? [#permalink] New post  29 Mar 2023, 04:52
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nicbo wrote:
Bunuel KarishmaB could you please explain this approach? Is there another approach that I can take?


The logic is discussed in detail here: https://anglesandarguments.com/blog-details/33
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Re: If 17!/7^m is an integer, what is the greatest possible value of m ? [#permalink] New post  29 Mar 2023, 07:37
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nicbo wrote:
Bunuel KarishmaB could you please explain this approach? Is there another approach that I can take?


You basically need to find the highest integer power of 7 in 17!.

For theory check: Number Properties

For questions check the following topics from our Special Questions Directory:

12. Trailing Zeros
13. Power of a number in a factorial

For more check Ultimate GMAT Quantitative Megathread

Hope it helps.
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Re: If 17!/7^m is an integer, what is the greatest possible value of m ? [#permalink] New post  29 Mar 2023, 09:14
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Bunuel wrote:
If 17!/7^m is an integer, what is the greatest possible value of m ?

A. 2
B. 3
C. 4
D. 5
E. 6


17! has two factors that are multiples of 7: 7 and 14 = 7^1 x 2.

Thus, we see that the maximum value of m is 2.

Answer: A
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Re: If 17!/7^m is an integer, what is the greatest possible value of m ? [#permalink]
29 Mar 2023, 09:14
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