2011 AMC 10B Problems/Problem 23: Difference between revisions
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==Problem== | ==Problem== | ||
What is the hundreds digit of 2011^2011? | What is the hundreds digit of <math>2011^{2011}</math>? | ||
(A) 1 (B) 3 (C) 4 (D) 6 (E) 8 | <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 21:51, 26 February 2011
Problem
What is the hundreds digit of 
?
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).
