In this paper, we prove the following: 1. Let G be a Moufang loop of order p(alpha)m, (p, m) = 1, (p-1, p(alpha)m,) = 1 and p is a prime. Suppose G has an element of order p(alpha). Then G = P x K, a split extension of a normal subloop K of order m with a subloop P order p(alpha). 2. Let G be a Moufang loop of odd order p(2)m, (p, m) = 1, and p is the smallest prime dividing \G\. Then a similar result holds as in (1) with alpha = 2.