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I'm not sure if there is a simpler way to approach this problem, but this is how I solved it. It relies on taking the remainder of each term individually, and from that, determining the remainder of the sum of the terms.

Remainder of 1/13 = 1 Remainder of 3/13 = 3 Remainder of (3^2)/13 = (Remainder of 3)*(Remainder of 3) / 13 = Remainder of 3*3/13 = 9 Remainder of (3^3)/13 = (Remainder of 3^2)*(Remainder of 3) / 13 = Remainder of 9*3/13 = 1 Remainder of (3^4)/13 = (Remainder of 3^3)*(Remainder of 3) / 13 = Remainder of 1*3/13 = 3

And so on....

Essentially, the question can be reduced to:

What is the remainder of: 1 + 3 + 9 + 1 + 3 + 9 + 1... (with 200 + 1 terms) when divided by 13

= (1+3+9) * (201/3) mod 13 (i.e. remainder) = 13 * (some integer) mod 13 = 0

I'm not sure if there is a simpler way to approach this problem, but this is how I solved it. It relies on taking the remainder of each term individually, and from that, determining the remainder of the sum of the terms.

Remainder of 1/13 = 1 Remainder of 3/13 = 3 Remainder of (3^2)/13 = (Remainder of 3)*(Remainder of 3) / 13 = Remainder of 3*3/13 = 9 Remainder of (3^3)/13 = (Remainder of 3^2)*(Remainder of 3) / 13 = Remainder of 9*3/13 = 1 Remainder of (3^4)/13 = (Remainder of 3^3)*(Remainder of 3) / 13 = Remainder of 1*3/13 = 3

And so on....

Essentially, the question can be reduced to:

What is the remainder of: 1 + 3 + 9 + 1 + 3 + 9 + 1... (with 200 + 1 terms) when divided by 13

= (1+3+9) * (201/3) mod 13 (i.e. remainder) = 13 * (some integer) mod 13 = 0

Therefore, the answer should be 0.

Good way to do it +1 Kudos
_________________

Thanks, Sri ------------------------------- keep uppp...ing the tempo...

Press +1 Kudos, if you think my post gave u a tiny tip

3^0/13= remainder 1. 3^0/13+3^1/13= remainder 4. 3^0/13+3^1/13+3^2/13= remainder 0. 3^3/13= remainder 1. 3^3/13+3^4/13= remainder 4. 3^3/13+3^4/13+3^5/13= remainder 0. . . . . Hence, sum of last 3 digits in the given equation must also give remainder 0. Some times, pattern of similar answers saves time, i guess.. works for me!!

If \(3^x\) is a number such that x is evenly divisible by 3 (ie. it leaves remainder of 0), then the sum of numbers from \(3^0\) to \(3^{x-1}\) will always be evenly divisible by 13.

Now, we know that 201 is divisible by 3. Therefore, 200 = 201 - 1 (which satisfies our condition)

Hence sum of the numbers from \(3^0\) to \(3^{200}\) will be divisible by 13.

Thus answer is 0.
_________________

Click below to check out some great tips and tricks to help you deal with problems on Remainders! http://gmatclub.com/forum/compilation-of-tips-and-tricks-to-deal-with-remainders-86714.html#p651942

Word Problems Made Easy! 1) Translating the English to Math : http://gmatclub.com/forum/word-problems-made-easy-87346.html 2) 'Work' Problems Made Easy : http://gmatclub.com/forum/work-word-problems-made-easy-87357.html 3) 'Distance/Speed/Time' Word Problems Made Easy : http://gmatclub.com/forum/distance-speed-time-word-problems-made-easy-87481.html

Is a Geometric progression having common ratio as '3' and number of terms as '201'.

Since Sum to n terms in GP = a(r^n-1)/(r-1)

where a=First term and r =common ration

Hence,

1*(3^201 -1 )/(3-1)

Rem of (3^201-1)/2 divided by 13

3^201 -1 /26

WKT, 3^3 = 27 = 26+1

{(26+1)^67 - 1}/26

{1-1}/26

=>0
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Rgds, TGC! _____________________________________________________________________ I Assisted You => KUDOS Please _____________________________________________________________________________

Re: The remainder when 1+3+3^2+3^3+..........+3^200 is divided 1 [#permalink]

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05 Oct 2014, 11:21

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Re: The remainder when 1+3+3^2+3^3+..........+3^200 is divided 1 [#permalink]

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13 Feb 2016, 06:54

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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