Difference between revisions of "2011 AMC 10B Problems/Problem 23"

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==Problem==  
 
==Problem==  
  
What is the hundreds digit of 2011^2011?
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What is the hundreds digit of <math>2011^{2011}</math>?
  
(A) 1 (B) 3 (C) 4 (D) 6 (E) 8
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<math>\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 8</math>
  
 
==Solution==
 
==Solution==

Revision as of 22:51, 26 February 2011

Problem

What is the hundreds digit of $2011^{2011}$?

$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 8$

Solution

(2000 + 11) ^ 2011 mod 1000 \n

11^2011 mod 1000

(10 + 1)^2011 mod 1000

2011C2 * 10^2 + 2011C1 * 10 + 1 mod 1000

500 + 110 + 1 mod 1000

611 mod 1000

So we know the last three digits of 2011 ^ 2011 is 611, and so the hundreds digit is 6 (D).

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