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Let k be a positive integer, and let G be a graph with minimum degree at least k. In their study (2010), Henning and Kazemi defined the k-tuple total domination number γ×k,tG of G as the minimum cardinality of a k-tuple total dominating set of G, which is a vertex set such that every vertex of G is adjacent to at least k vertices in it. If G̅ is the complement of G, the complementary prism GG̅ of G is the graph formed from the disjoint union of G and G̅ by adding the edges of a perfect matching between the corresponding vertices of G and G̅. In this paper, we extend some of the results of Haynes et al. (2009) for the k-tuple total domination number and also obtain some other new results. Also we find the k-tuple total domination number of the complementary prism of a cycle, a path, or a complete multipartite graph.