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05 Nov 2009, 11:52
Kudos
Bookmarks
C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
75%
(hard)
Question Stats:
61% (02:24) correct
39%
(02:38)
wrong
based on 411
sessions
History
Date
Time
Result
Not Attempted Yet
What is the remainder when \(1 + 3 + 3^2 + 3^3 + ... + 3^{200}\) is divided 13 ?
A. 12
B. 7
C. 0
D. 5
E. 3
A. 12
B. 7
C. 0
D. 5
E. 3
05 Nov 2009, 13:21
Kudos
Bookmarks
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.
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.
General Discussion
05 Nov 2009, 13:54
Kudos
Bookmarks
AKProdigy87
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.
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
but was a bit confused between 200 and 201 terms 
