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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8027
GMAT 1: 760 Q51 V42 GPA: 3.82
a0 = 0, a1 = 1. an is the remainder when an-1 + an-2 is divided by 3 f  [#permalink]

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Difficulty:   75% (hard)

Question Stats: 43% (03:09) correct 57% (03:01) wrong based on 23 sessions

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[GMAT math practice question]

a0 = 0, a1 = 1. an is the remainder when an-1 + an-2 is divided by $$3$$ for $$n ≥ 2$$. What is the value of a101 + a102 + a103 + a104 + a105 + a106 + a107 + a108?

A. 0
B. 3
C. 5
D. 7
E. 9

_________________
Senior Manager  P
Joined: 09 Jun 2014
Posts: 352
Location: India
Concentration: General Management, Operations
a0 = 0, a1 = 1. an is the remainder when an-1 + an-2 is divided by 3 f  [#permalink]

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MathRevolution wrote:
[GMAT math practice question]

a0 = 0, a1 = 1. an is the remainder when an-1 + an-2 is divided by $$3$$ for $$n ≥ 2$$. What is the value of a101 + a102 + a103 + a104 + a105 + a106 + a107 + a108?

A. 0
B. 3
C. 5
D. 7
E. 9

This is cyclicity question :

The series is as below
A1- 1
A2- 1
A3- 2
A4- 0
A5- 2
A-6 2
A7- 1
A8- 0

So together the sum of first segments gives 9 ..Now picking up any random segments will lead to same set of digits which will sum up to 9

So for even A161+ A162+ ...A168 =sum of digits will be 9 .
or A3 to A11 Hope it helps!!

Originally posted by prabsahi on 08 Feb 2019, 03:52.
Last edited by prabsahi on 08 Feb 2019, 04:51, edited 2 times in total.
Director  G
Joined: 09 Mar 2018
Posts: 994
Location: India
Re: a0 = 0, a1 = 1. an is the remainder when an-1 + an-2 is divided by 3 f  [#permalink]

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1
MathRevolution wrote:
[GMAT math practice question]

a0 = 0, a1 = 1. an is the remainder when an-1 + an-2 is divided by $$3$$ for $$n ≥ 2$$. What is the value of a101 + a102 + a103 + a104 + a105 + a106 + a107 + a108?

A. 0
B. 3
C. 5
D. 7
E. 9

The trick was to find a pattern.
Keyword = Remainder when divided by 3
an is the remainder when an-1 + an-2 is divided by $$3$$

a_0 = 0
a_1 = 1
a_2 = 1 (1+0)/3
a_3 = 2 (1+1)/3
a_4 = 0 (2+1)/3
a_5 = 2 (0+2)/3
a_6 = 2 (0+2)/3
a_7 = 1 (2+2)/3
a_8 = 0(2+1)/3

If you go forward you will notice the same series.

So now the series starts from 101 ...... 108
1+1+2+0+2+2+1+0
9

E
_________________
If you notice any discrepancy in my reasoning, please let me know. Lets improve together.

Quote which i can relate to.
Many of life's failures happen with people who do not realize how close they were to success when they gave up.
Senior Manager  P
Joined: 09 Jun 2014
Posts: 352
Location: India
Concentration: General Management, Operations
Re: a0 = 0, a1 = 1. an is the remainder when an-1 + an-2 is divided by 3 f  [#permalink]

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KanishkM wrote:
MathRevolution wrote:
[GMAT math practice question]

a0 = 0, a1 = 1. an is the remainder when an-1 + an-2 is divided by $$3$$ for $$n ≥ 2$$. What is the value of a101 + a102 + a103 + a104 + a105 + a106 + a107 + a108?

A. 0
B. 3
C. 5
D. 7
E. 9

The trick was to find a pattern.
Keyword = Remainder when divided by 3
an is the remainder when an-1 + an-2 is divided by $$3$$

a_0 = 0
a_1 = 1
a_2 = 1 (1+0)/3
a_3 = 2 (1+1)/3
a_4 = 0 (2+1)/3
a_5 = 2 (0+2)/3
a_6 = 2 (0+2)/3
a_7 = 1 (2+2)/3
a_8 = 0(2+1)/3

If you go forward you will notice the same series.

So now the series starts from 101 ...... 108
1+1+2+0+2+2+1+0
9

E

I see .I made a mistake Thanks
Director  G
Joined: 09 Mar 2018
Posts: 994
Location: India
Re: a0 = 0, a1 = 1. an is the remainder when an-1 + an-2 is divided by 3 f  [#permalink]

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prabsahi wrote:

I see .I made a mistake Thanks

prabsahi, your approach was right. But i still feel that is not the easiest way to solve it. _________________
If you notice any discrepancy in my reasoning, please let me know. Lets improve together.

Quote which i can relate to.
Many of life's failures happen with people who do not realize how close they were to success when they gave up.
Senior Manager  P
Joined: 09 Jun 2014
Posts: 352
Location: India
Concentration: General Management, Operations
Re: a0 = 0, a1 = 1. an is the remainder when an-1 + an-2 is divided by 3 f  [#permalink]

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KanishkM wrote:
prabsahi wrote:

I see .I made a mistake Thanks

prabsahi, your approach was right. But i still feel that is not the easiest way to solve it. yes..You are right KanishkM..Will update it Thanks
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Joined: 18 Aug 2017
Posts: 5032
Location: India
Concentration: Sustainability, Marketing
GPA: 4
WE: Marketing (Energy and Utilities)
Re: a0 = 0, a1 = 1. an is the remainder when an-1 + an-2 is divided by 3 f  [#permalink]

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MathRevolution wrote:
[GMAT math practice question]

a0 = 0, a1 = 1. an is the remainder when an-1 + an-2 is divided by $$3$$ for $$n ≥ 2$$. What is the value of a101 + a102 + a103 + a104 + a105 + a106 + a107 + a108?

A. 0
B. 3
C. 5
D. 7
E. 9

the pattern here followed is of divisibility of a no with 3 ; which is always for consective no is series of 0,1,2,0,1,2
0,1,2 is our pattern
so the 99th term would be 2 and 101st would be 1
so sum = 1+2+0+1+2+0+1+2; 9
IMO E
Senior Manager  P
Joined: 09 Jun 2014
Posts: 352
Location: India
Concentration: General Management, Operations
a0 = 0, a1 = 1. an is the remainder when an-1 + an-2 is divided by 3 f  [#permalink]

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Archit3110 wrote:
MathRevolution wrote:
[GMAT math practice question]

a0 = 0, a1 = 1. an is the remainder when an-1 + an-2 is divided by $$3$$ for $$n ≥ 2$$. What is the value of a101 + a102 + a103 + a104 + a105 + a106 + a107 + a108?

A. 0
B. 3
C. 5
D. 7
E. 9

the pattern here followed is of divisibility of a no with 3 ; which is always for consective no is series of 0,1,2,0,1,2
0,1,2 is our pattern
so the 99th term would be 2 and 101st would be 1
so sum = 1+2+0+1+2+0+1+2; 9
IMO E

Hi Archit,

I guess you made a similar mistake like I did.

Please check the pattern mentioned in my previous post or KanishkM's.I have corrected it.

Its 0 1 1 2 0 2 2 1 0

Hope it helps !!
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8027
GMAT 1: 760 Q51 V42 GPA: 3.82
Re: a0 = 0, a1 = 1. an is the remainder when an-1 + an-2 is divided by 3 f  [#permalink]

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=>

a0 = 0
a1 = 1
a2 = a1 + a0 = 1 + 0 = 1
a3 = a2 + a1 = 1 + 1 = 2
a4 = 0 since a3 + a2 = 2 + 1 = 3 = 3(1)+0, and the remainder when a3 + a2 is divided by 3 is zero.
a5 = a4 + a3 = 0 + 2 = 2
a6 = a5 + a4 = 2 + 0 = 2
a7 = 1 since a6 + a5 = 2 + 2 = 4= 3(1)+1, and the remainder when a6 + a5 is divided by 3 is 1.
a8 = 0 since a7 + a6 = 1 + 2 = 3= 3(1)+0, the remainder is 0, when a7 + a6 is divided by 3.
a9 = a8 + a7 = 0 + 1 = 1
Thus, the sequence is periodic, with period 8.
a101 + a102 + a103 + a104 + a105 + a106 + a107 + a108
= a5 + a6 + a7 + a0 + a1 + a2 + a3 + a4
$$= 2 + 2 + 1 + 0 + 1 + 1 + 2 + 0 = 9$$
Therefore, the answer is E.
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Joined: 13 Mar 2018
Posts: 14
Re: a0 = 0, a1 = 1. an is the remainder when an-1 + an-2 is divided by 3 f  [#permalink]

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KanishkM wrote:
MathRevolution wrote:
[GMAT math practice question]

a0 = 0, a1 = 1. an is the remainder when an-1 + an-2 is divided by $$3$$ for $$n ≥ 2$$. What is the value of a101 + a102 + a103 + a104 + a105 + a106 + a107 + a108?

A. 0
B. 3
C. 5
D. 7
E. 9

The trick was to find a pattern.
Keyword = Remainder when divided by 3
an is the remainder when an-1 + an-2 is divided by $$3$$

a_0 = 0
a_1 = 1
a_2 = 1 (1+0)/3
a_3 = 2 (1+1)/3
a_4 = 0 (2+1)/3
a_5 = 2 (0+2)/3
a_6 = 2 (0+2)/3
a_7 = 1 (2+2)/3
a_8 = 0(2+1)/3

If you go forward you will notice the same series.

So now the series starts from 101 ...... 108
1+1+2+0+2+2+1+0
9

E

I dont understand what quotient to take here. Please explain. Re: a0 = 0, a1 = 1. an is the remainder when an-1 + an-2 is divided by 3 f   [#permalink] 13 Feb 2019, 07:51
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