Let A be a nonempty finite set of positive integers. Suppose that for any two elements

Chapter 0, Problem 19

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Let A be a nonempty finite set of positive integers. Suppose that for any two elements r; s 2 A, we have rjs or sjr. (In symbols, 8r 2 A; 8s 2 A; .rjs or sjr/.) a. Prove that A contains an element t with the property that for all a 2 A, ajt. (In symbols, 9t 2 A; 8a 2 A; ajt.) b. Furthermore, prove that t is unique (i.e., there is only one element of A that is a multiple of all elements of A). c. Finally, give an example to show that uniqueness does not hold if we do not assume that all the elements of A are positive.