A Festival of Divisibility

Most people know a few "rules for divisibility".  Things like, "If a number ends in a five or a zero, it is divisible by 5."  But why is that rule true?  Why can we focus only on the right-most digit, and ignore all the others?

Some other rules are even more challenging and interesting.  So we turned this question into an I Love Math problem for week 3.  Here it is:

This last challenge is the most interesting.  It requires the most deep thinking about place value.  It forces children (and all of us) to consider "why" a rule works, and contemplate the structure of our number system.