I'll give it a try Yaeger- and hopefully someone more senior on the board will fix up the misakes. I guess you know how the mod function works...

We'll do it with mod 3 because that's easier:

Notice how (2 mod 3) = (5 mod 3) = 2. Well, when you think about mod 3 as being a relation on the set of whole numbers, there are infinitely many numbers (every 3rd member of the Set to be exact) that are related to 2 by the mod 3 relation. So when you start to write down the members of the relationship, you get:

An equivalence class is a short-hand way of naming the three partitions of the relationship, that is all the members of the relationship that have either (x, 0), (y, 1) or (z, 3) in them, so are two equivalence classes which are congruent. Both of them are elements from the same representational system, or partition.

Another example of equivalence classes are in the rational numbers, where

Question 2 just shows you how you can do some basic math stuff with them. Try it out.