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

(Solution)
(Solution)
Line 9: Line 9:
 
(2000  + 11) ^ 2011 mod 1000 \n
 
(2000  + 11) ^ 2011 mod 1000 \n
  
11^2011 mod 1000 \n
+
11^2011 mod 1000  
(10 + 1)^2011 mod 1000 \n
+
 
2011C2 * 10^2 + 2011C1 * 10 + 1    mod 1000 \n
+
(10 + 1)^2011 mod 1000  
500 + 110 + 1  mod 1000 \n
+
 
611 mod 1000 \n
+
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).

Invalid username
Login to AoPS