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 *…') |
Epicfailiure (talk | contribs) (→Solution) |
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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).
