1) Consider the following language: L = {w | w {a,b}* and w contains the following properties} Property 1: w contains an odd number of as or an odd number of bs, but not both. Property 2: every b in w is immediately followed by at least one a. Design a minimal DFA that accepts L. Label each state with its the shortlex class representative.

2) Consider the general case of a DFA that recognizes the set of strings representing base-B unsigned integers, read MSB-first, that are divisible by D. Develop a concise, formal description of this machine in terms of the parameters B and D. Your machine must NOT accept the empty string. 3) Consider the following claim: A string of 0s and 1s that represents a base-2 value that is congruent to 2 modulo-5 cannot also represent a base-5 value that evenly divisible by 3. Design a suitable minimal DFA that proves whether this claim is true or not. If it is not true, what is the shortest (in a shortlex sense) string that serves as a counter example. Verify that this counterexample is valid by determining the integer represented by the string (expressed in base-10) under each interpretation and what its residue is for each modulus.