In class we discussed a "shift" encryption scheme which arranges the letters A through Z in a circle and uses rotation (by a secret number of letters) as a means for encryption. The basic algebraic situation here appears in many number-theoretic encryption schemes. Consider the set of integers, which we will denote Z = . . . ,-2,-1, 0, 1, 2, . . . . Fix a particular nonzero integer n (which, for the moment, you can think of as 26). (a) We say that two numbers a and b are congruent modulo n if n evenly divides their difference. The conventional notation for this is a b modn or, more compactly, a b Symbolically, a-n b exactly when a-b = kn for some integer k. Prove that if a-n b then (b) Prove that for any integer a, it is the case that a-n a. (You would say that the relation is (c) Prove that if a-n b and b=n c, then a =n e. (You would say that the relation-n is transitive.) A relation that satisfies these properties is called an equivalence relation. You will have learned about b-n a. (You would say that the relation is symmetric.) reflerive.) these in your Discrete Math class. If you would like a review, read Chapter 11 of R. Hammack's excellent (and free) book The Book of Proof