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The remainder when 10^2006+2006 divided by 9 is; A)0 B)2 C)5 D)6 E)7

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christiano-cr7 | Salutatorian

Posted July 18, 2013 at 5:04 PM via web

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The remainder when 10^2006+2006 divided by 9 is;

A)0

B)2

C)5

D)6

E)7

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embizze | High School Teacher | (Level 1) Educator Emeritus

Posted July 19, 2013 at 2:07 AM (Answer #2)

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What is the remainder for `(10^2006 +2006)/9` ?

`(10^2006 +2006)/9=10^2006/9+2006/9`

Now a power of 10 divided by 9 always has a remainder of 1. ``

9*1+1=10

9*11+1=100

9*111+1=1000 etc...

So we have 1 plus the remainder when 2006 is divided by 9 which is 8.

The sum of 2 numbers will have the same remainder as the sum of the remainders.

So the remainder would be 9 -- but 9 is a factor of 9.

The remainder is 0 -- (A) is the answer.

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embizze | High School Teacher | (Level 1) Educator Emeritus

Posted July 19, 2013 at 2:00 AM (Reply #1)

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But `(9+1)^2006 != 9^2006+1^2006`

 You can use the binomial expansion to show that the missing terms are all products of some power of 9 -- then your conclusion holds.

 

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