For an irreducible chain, demonstrate that aperiodicity is a necessary and sufficient condition for some power P n of the transition matrix P to have all entries positive. (Hint: For sufficiency, you may use the following number theoretic fact: Suppose S is a set of positive integers that is closed under addition and has greatest common divisor 1.

Then there exists an integer m such that n ∈ S whenever n ≥ m.)