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What is the greatest positive integer n such that 5^n divides 10! – (2
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parkhydel
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What is the greatest positive integer n such that 5^n divides 10! – (2 [#permalink] New post  27 Apr 2020, 14:46
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What is the greatest positive integer n such that \(5^n\) divides \(10! – (2)(5!)^2\) ?

A. 2
B. 3
C. 4
D. 5
E. 6



PS14051.02
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nick1816
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What is the greatest positive integer n such that 5^n divides 10! – (2 [#permalink] New post  27 Apr 2020, 15:43
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\(10!- 2(5!)^2\) = \(5! [(10*9*8*7*6)- (2*5!)]\)

= \(5! [(2*5*3*3*2*4*7*2*3)- (2*5!)]\)

= \(5! [(2*5*4*3*2*2*7*3*3)- (2*5!)]\)

= \(5![(2*5!*126) - 2(5!)]\)

=\(2*5!*5!(126-1)\)

=\(2*5!*5!*5^3\)

= \(2*4!*5*4!*5*5^3\)

n= 1+1+3=5


parkhydel wrote:
What is the greatest positive integer n such that \(5^n\) divides \(10! – (2)(5!)^2\) ?

A. 2
B. 3
C. 4
D. 5
E. 6



PS14051.02
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lacktutor
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Re: What is the greatest positive integer n such that 5^n divides 10! – (2 [#permalink] New post  27 Apr 2020, 17:30
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parkhydel wrote:
What is the greatest positive integer n such that \(5^n\) divides \(10! – (2)(5!)^2\) ?

A. 2
B. 3
C. 4
D. 5
E. 6



PS14051.02


\(10! = 5!* 6*7*8*9*10 = 5!*(2*3)*7*(2*4)*9(2*5)= (5!)^{2}* 252\)

--> \(10! – (2)(5!)^2= (5!)^{2}* 252 - (2)(5!)^2= (5!)^2 (252-2)= 250*(5!)^2= 2*5^{3}*(5!)^{2}\)
\(5^{n} =5^{5}\)
--> \(n =5 \)

Answer (D).
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chetan2u
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What is the greatest positive integer n such that 5^n divides 10! – (2 [#permalink] New post  27 Apr 2020, 19:08
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parkhydel wrote:
What is the greatest positive integer n such that \(5^n\) divides \(10! – (2)(5!)^2\) ?

A. 2
B. 3
C. 4
D. 5
E. 6



PS14051.02


\(10! – (2)(5!)^2=5!(10*9*8*7*6-2*5!)\)

\(5!(2*5*3*3*2*2*2*7*2*3-2*2*3*2*2*5)=5!(2^5*3^3*5*7-2^4*3*5)=\)

\(=5!*2^4*3*5(2*3^2*7-1)=5!*2^4*3*5(126-1)=\)

\(5*4!*2^4*3*5*125=2^4*3*5^{1+1+3}*4!\)

So we have five 5s in the terms, and the maximum value of n is 5

D
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html
4) Base while finding % increase and % decrease : https://gmatclub.com/forum/percentage-increase-decrease-what-should-be-the-denominator-287528.html


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What is the greatest positive integer n such that 5^n divides 10! – (2 [#permalink] New post  27 Apr 2020, 19:23
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parkhydel wrote:
What is the greatest positive integer n such that \(5^n\) divides \(10! – (2)(5!)^2\) ?

A. 2
B. 3
C. 4
D. 5
E. 6



PS14051.02


\(10! – (2)(5!)^2 = 5!*(6*7*8*9*10 - 2*120) = 5!*240*(7*2*9-1) = 5^2*24*48*(125) = 5^5*24*48\)



i.e. if \(5^n\) is divisor of \(10!-2*5!^2\) then,

\(n_{max} = 5 \)



Answer: Option D
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Louis14
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What is the greatest positive integer n such that 5^n divides 10! – (2 [#permalink] New post  07 Nov 2020, 10:54
10! - 2(5!)(5!)

10*9*8*7*6*5! - 2(5!)(5!)

5! [10*9*8*7*6 - 2(5*4*3*2)]

5! [10*9*8*7*6 - 10*8*3]

5! [80 (9*7*6) - 80(3)]

5! [80(375)]

Now, 5! has one 5; 80 has one 5; 375 has three 5s.

So, the max number of 5s are 5 i.e. n = 5

Kudos = Thank you. :)
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