Solution:
1

Explanation:
This one is so sneaky.  

First, consider 100 divided by 11.  The remainder here is 1.  Now consider the remainder when 100x100 is divided by 11.  Don't do it on your calculator or on paper.  Rather, consider that you have one hundred hundreds, and each of them has a remainder of 1 when divided by 11.  So, go through each of your hundred hundreds and divide it by 11, leaving remainder 1.  Then collect up your remainders into a single hundred, and divide it by 11, leaving a remainder of 1.  This process can be extended to dividing 100x100x100 by 11, and indeed, to dividing any power of 100 by 11.