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

(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 *…')
 
(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).

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