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Letters of the word 'ARRANGEMENT' are first written in ascending order

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Letters of the word 'ARRANGEMENT' are first written in ascending order  [#permalink]

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New post 26 Dec 2019, 06:49
2
8
00:00
A
B
C
D
E

Difficulty:

  75% (hard)

Question Stats:

29% (01:43) correct 71% (02:20) wrong based on 31 sessions

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Q. Letters of the word 'ARRANGEMENT' are first written in ascending order and then in descending order, and this process is continued. The 8^(12)th letter of the above series is:

A. A
B. R
C. E
D. N
E. T
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Re: Letters of the word 'ARRANGEMENT' are first written in ascending order  [#permalink]

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New post 26 Dec 2019, 14:21
2
2
There are 11 letters in ARRANGEMENT; there will be 22 letters in 1 cycle.

We need to find the remainder when 8^{12} or 2^{36} is divided by 22.

22=2*11

\(2^{36}\)= 0 mod 2

\(2^{5}\)= -1 mod 11

\([2^5]^7\)= (-1)^7 mod 11

\([2^5]^7\)= (-1) mod 11

\([2^{35}]\)= (-1) mod 11

\([2^{35}]*2\)= (-1*2) mod 11

\(2^{36}\)= -2 mod 11= 9 mod 11

\(2^{36}\)= LCM(2,11)k+20

\(2^{36}\)= 22k+20

Hence 8^{12}th word is 20th word of the first cycle, that is E.






uchihaitachi wrote:
Q. Letters of the word 'ARRANGEMENT' are first written in ascending order and then in descending order, and this process is continued. The 8^(12)th letter of the above series is:

A. A
B. R
C. E
D. N
E. T
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Re: Letters of the word 'ARRANGEMENT' are first written in ascending order  [#permalink]

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New post 10 Jan 2020, 01:15
nick1816

Ascending order and then discending order would lead to this cycle:

ARRANGEMENT-TNEMEGNARRA --> 20th letter is R

Where am I wrong?
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Re: Letters of the word 'ARRANGEMENT' are first written in ascending order  [#permalink]

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New post 10 Jan 2020, 02:07
It's a very good question; Poorly written tho.

You gotta write the 'Arrangement' in alphabetical order in the case of ascending order and in reverse of alphabetical order in the case of descending order.



Camach700 wrote:
nick1816

Ascending order and then discending order would lead to this cycle:

ARRANGEMENT-TNEMEGNARRA --> 20th letter is R

Where am I wrong?
Intern
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Joined: 16 Dec 2019
Posts: 18
Re: Letters of the word 'ARRANGEMENT' are first written in ascending order  [#permalink]

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New post 22 Jan 2020, 07:51
nick1816 wrote:
There are 11 letters in ARRANGEMENT; there will be 22 letters in 1 cycle.

We need to find the remainder when 8^{12} or 2^{36} is divided by 22.

22=2*11

\(2^{36}\)= 0 mod 2

\(2^{5}\)= -1 mod 11

\([2^5]^7\)= (-1)^7 mod 11

\([2^5]^7\)= (-1) mod 11

\([2^{35}]\)= (-1) mod 11

\([2^{35}]*2\)= (-1*2) mod 11

\(2^{36}\)= -2 mod 11= 9 mod 11

\(2^{36}\)= LCM(2,11)k+20

\(2^{36}\)= 22k+20

Hence 8^{12}th word is 20th word of the first cycle, that is E.






uchihaitachi wrote:
Q. Letters of the word 'ARRANGEMENT' are first written in ascending order and then in descending order, and this process is continued. The 8^(12)th letter of the above series is:

A. A
B. R
C. E
D. N
E. T



Can you explain this without using mod ?
Thanks
VP
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Posts: 1314
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Re: Letters of the word 'ARRANGEMENT' are first written in ascending order  [#permalink]

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New post 22 Jan 2020, 10:03
1
22 = 2*11

\(2^{36}\) when divided by 2 leaves 0 remainder.



\(2^5 = (33-1)\)

\((2^5)^7 = (33-1)^7\)

\((2^5)^7= (33-1)^7\) = Rem (-1)^7 / 11 = Rem (-1) / 11

\((2^5)^7*2\)= Rem(-1*2) / 11

\(2^{36}\) when divided by 11 leaves -2 or (11-2=9) remainder.

So basically our question stem is The remainder\( 2^{36}\) when divided by 2 is 0, and when divided by 11 is 9. What is the remainder when \(2^{36}\) is divided by 22.

LCM (2, 11)= 22

Find the numbers less than 22 when divided by 11 leaves 9 remainder.

9 and 20

only 20 when divided by 2 leaves 0 remainder.

hence \(2^{36}\) when divided by 22 leaves 20 remainder.







allkagupta wrote:
nick1816 wrote:
There are 11 letters in ARRANGEMENT; there will be 22 letters in 1 cycle.

We need to find the remainder when 8^{12} or 2^{36} is divided by 22.

22=2*11

\(2^{36}\)= 0 mod 2

\(2^{5}\)= -1 mod 11

\([2^5]^7\)= (-1)^7 mod 11

\([2^5]^7\)= (-1) mod 11

\([2^{35}]\)= (-1) mod 11

\([2^{35}]*2\)= (-1*2) mod 11

\(2^{36}\)= -2 mod 11= 9 mod 11

\(2^{36}\)= LCM(2,11)k+20

\(2^{36}\)= 22k+20

Hence 8^{12}th word is 20th word of the first cycle, that is E.






uchihaitachi wrote:
Q. Letters of the word 'ARRANGEMENT' are first written in ascending order and then in descending order, and this process is continued. The 8^(12)th letter of the above series is:

A. A
B. R
C. E
D. N
E. T



Can you explain this without using mod ?
Thanks
GMAT Club Bot
Re: Letters of the word 'ARRANGEMENT' are first written in ascending order   [#permalink] 22 Jan 2020, 10:03
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Letters of the word 'ARRANGEMENT' are first written in ascending order

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