a. Let n be a positive integer and consider B to be the

Chapter 8, Problem 30

(choose chapter or problem)

a. Let n be a positive integer and consider B to be the set of all positive integer divisors of n. Prove that (B, d) is a partially ordered set where x d y means x 0 y. In the terminology of Exercise 29, the least upper bound of x and y is the least common multiple of x and y, and the greatest lower bound is the greatest common divisor. (B, d) is a distributive lattice. b. Prove that for n = 6, (B, d) is a Boolean algebra. (Hint: 1 is the least element and 6 is the greatest element). c. For n = 8, (B, d) is not a Boolean algebra. (i) Show that this is true by using the definition of a Boolean algebra. (ii) Show that this is true by using Exercise 6.