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How many trailing zeros will 11^50 - 1 will have ?

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Intern
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Joined: 28 Mar 2015
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How many trailing zeros will 11^50 - 1 will have ?  [#permalink]

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New post 28 Mar 2015, 21:39
1
6
00:00
A
B
C
D
E

Difficulty:

  75% (hard)

Question Stats:

47% (01:04) correct 53% (01:48) wrong based on 51 sessions

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How many trailing zeros will 11^50 - 1 will have ?

A) 3
B) 4
C) 5
D) 6
E) 7
Director
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Joined: 07 Aug 2011
Posts: 535
Concentration: International Business, Technology
GMAT 1: 630 Q49 V27
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Re: How many trailing zeros will 11^50 - 1 will have ?  [#permalink]

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New post 29 Mar 2015, 03:25
CrazyIvan wrote:
How many trailing zeros will \(11^5^0 - 1\) will have ?

A) 3
B) 4
C) 5
D) 6
E) 7


i doubt this is a GMAT standard question , but let try to give it a shot .

\(11^5^0 - 1= (1+10)^50 -1 = 1+ ^5^0C_1 *10 + ^5^0C_2 *10^2 + ^5^0C_3 *10^3 + .... +^5^0C_50 *10^5^0 -1\)

\(50*10 (1+ \frac{49}{2*1}*10 + \frac{49*48}{3*2*1} * 10^2 + ..... + \frac{49*48...*2}{50!} 10^4^9)\)

taking LCM 50!

\(50*10 *( 50!+ \frac{49}{2*1}*10 *\frac{50!}{2!} + \frac{49*48}{3*2*1} * 10^2 *\frac{50!}{3!} + .....)/50!\)

we can take out 50 from the inner brackets and the remaining quantity will still be > 1

so answer should be 3 .

but i am not very sure .
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Re: How many trailing zeros will 11^50 - 1 will have ?  [#permalink]

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New post 29 Mar 2015, 09:28
It took me a lot of time to solve it, not only 2 minutes... so I wouldn't have been able to solve it in a test

I noticed that the powers of 11 can be calculated as
\(11^0=1 \\11^1=11 \\ 11^2=121 \\11^3=1331 \\11^4=14641\)

It reminds me of the Pascal's triangle, and the powers of 11 can be written as \(11^n=\sum_{i=0}^{n} (10^{i} \cdot \frac{n!}{i!(n-i)!})\)

I consider the last 4 digits, so I calculate \(1000\cdot \frac{50!}{47!3!}+100\cdot \frac{50!}{48!2!}+10 \cdot \frac{50!}{49!1!} +1=39200000+1225+500+1=39223001\)

If we subtract 1, there are three zeroes, so A is the correct answer.

Unfortunately I cannot explain it in a simpler way :(
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Re: How many trailing zeros will 11^50 - 1 will have ?  [#permalink]

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New post 12 Jul 2018, 06:05
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Re: How many trailing zeros will 11^50 - 1 will have ? &nbs [#permalink] 12 Jul 2018, 06:05
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