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

Epicfailiure (talk | contribs) (Created page with '==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 11^2011 mod 1000 (10 + 1)^2011 mod 1000 2011C2 *…') |
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==Solution== | ==Solution== | ||

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

− | 11^2011 mod 1000 | + | |

− | (10 + 1)^2011 mod 1000 | + | 11^2011 mod 1000 \n |

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

− | 500 + 110 + 1 mod 1000 | + | 2011C2 * 10^2 + 2011C1 * 10 + 1 mod 1000 \n |

− | 611 mod 1000 | + | 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). | So we know the last three digits of 2011 ^ 2011 is 611, and so the hundreds digit is 6 (D). |

## Revision as of 20:41, 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 \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).