2011 AMC 10B Problems/Problem 23

Problem

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

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

(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).