2011 AMC 10B Problem 23

Question

What is the hundreds digit of

2011^{2011}?

1

4

5

6

9

Work modulo

1000

In

(10+1)^{2011}

, terms with

10^3

or higher do not affect the last three digits

Because

2011\equiv11\pmod{1000}

, it is enough to find

11^{2011}=(10+1)^{2011}

modulo

1000

.

All terms with

10^3

or higher are divisible by

1000

, so only the first three terms matter:

1+2011\cdot10+\binom{2011}{2}10^2.

Modulo

1000

, this is

\begin{aligned} &1+11\cdot10+\dfrac{2011\cdot2010}{2}\cdot100 \\ &\quad \equiv1+110+500=611 \end{aligned}

.

The hundreds digit is therefore

6

.

Thus, D is the correct answer.

Problem 23 in other years