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10! - 2(5!)^2 is divisible by 10^n

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10! - 2(5!)^2 is divisible by 10^n [#permalink] New post   12 Aug 2021, 07:27
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\(10! - 2(5!)^2\) is divisible by \(10^n\)

Quantity A
Quantity B
Greatest integer value of \(n\)
5


A. Quantity A is greater
B. Quantity B is greater
C. The two quantities are equal
D. The relationship cannot be determined from the information given
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C
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Re: 10! - 2(5!)^2 is divisible by 10^n [#permalink] New post   13 Aug 2021, 21:26
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COolguy101 wrote:
Pujamahadik wrote:
5!5! *2(126-1) =5!5!*125 = 10^5 *36
C


How much time do you spend on this question? It took me almost four minutes :lol: :lol: :lol:


I would say it will take under a minute if your concepts of exponents are sound :thumbsup:

\(10! - 2(5!)^2\)

\((10)(9)(8)(7)(6)(5!) - 2(5!)(5!)\)
\(5![(10)(9)(8)(7)(6) - 2(5!)]\)
\(5![30240 - 2(120)]\)
\(120[30240 - 240]\)
\(120(30000)\)
\(3600000 = 36(10^5)\)
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10! - 2(5!)^2 is divisible by 10^n [#permalink] New post   12 Aug 2021, 08:16
5!5! *2(126-1) =5!5!*125 = 10^5 *36
C
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Re: 10! - 2(5!)^2 is divisible by 10^n [#permalink] New post   12 Aug 2021, 23:52
Pujamahadik wrote:
5!5! *2(126-1) =5!5!*125 = 10^5 *36
C



How much time do you spend on this question? It took me almost four minutes :lol: :lol: :lol:
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10! - 2(5!)^2 is divisible by 10^n [#permalink] New post   13 Aug 2021, 01:52
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a number is divisible by 10 if it has factors 5 and 2
10!=10*....5*...2 has \(10^n\) with the n=2
\((5!)^2\) has (5*...2) raised to the power of 2 results in n=2
\(10^2(9*8*7*6*4*3-2*16*9)=32*9*10^2(7*6*3-1)=32*9*10^2*125\) ---> contains \(2^5\) and \(5^3\) besides \(10^2\). The number n=5. Answer is C.

:) this question resembles me one of the very hard ones from magoosh and would take me 3+ min. of the solution time


KarunMendiratta wrote:
\(10! - 2(5!)^2\) is divisible by \(10^n\)

Quantity A
Quantity B
Greatest integer value of \(n\)
5


A. Quantity A is greater
B. Quantity B is greater
C. The two quantities are equal
D. The relationship cannot be determined from the information given
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10! - 2(5!)^2 is divisible by 10^n [#permalink] New post   13 Aug 2021, 19:03
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Question says 'is divisible by'. Technically, it is divisible by 10^6.
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Re: 10! - 2(5!)^2 is divisible by 10^n [#permalink] New post   06 Dec 2025, 07:50
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Re: 10! - 2(5!)^2 is divisible by 10^n [#permalink]
06 Dec 2025, 07:50
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