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

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(2000 + 11) ^ 2011 mod 1000 \n | (2000 + 11) ^ 2011 mod 1000 \n | ||

− | 11^2011 mod 1000 | + | 11^2011 mod 1000 |

− | (10 + 1)^2011 mod 1000 | + | |

− | 2011C2 * 10^2 + 2011C1 * 10 + 1 mod 1000 | + | (10 + 1)^2011 mod 1000 |

− | 500 + 110 + 1 mod 1000 | + | |

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

## Revision as of 20:42, 25 February 2011

## 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

(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).