Problem:
What is the hundreds digit of 20112011?
Answer Choices:
A. 1
B. 4
C. 5
D. 6
E. 9
Solution:
In the expansion of (2000+11)2011, all terms except 112011 are divisible by 1000 , so the hundreds digit of 20112011 is equal to that of 112011. Furthermore, in the expansion of (10+1)2011, all terms except 12011, (12011)(10)(12010), and (22011)(10)2(12009) are divisible by 1000 . Thus the hundreds digit of 20112011 is equal to that of
1+(20111)(10)(12010)+(20112)(10)2(12009)=1+2011⋅10+2011⋅1005⋅100=1+2011⋅100510.
Finally, the hundreds digit of this number is equal to that of 1+11⋅510=5611, so the requested hundreds digit is (D)6.
The problems on this page are the property of the MAA's American Mathematics Competitions