2011 AMC 10B Problems/Problem 23

Problem

What is the hundreds digit of 2011^2011?

(A) 1 (B) 3 (C) 4 (D) 6 (E) 8

Solution

(2000 + 11) ^ 2011 mod 1000 \n

11^2011 mod 1000 \n (10 + 1)^2011 mod 1000 \n 2011C2 * 10^2 + 2011C1 * 10 + 1 mod 1000 \n 500 + 110 + 1 mod 1000 \n 611 mod 1000 \n

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