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Elementary flux modes are an important class of metabolic pathways used to characterize the functioning and behavior of metabolic networks of biochemical reactions in a biological cell. The computation of the elementary flux modes is accomplished by using the so-called Nullspace Algorithm whose high computational cost and memory requirements still limit the computation to relatively small metabolic networks. We combine a "combinatorial'' parallelization with a novel divide-and-conquer paradigm into a new implementation of the Nullspace Algorithm with lower memory requirements. We discuss the disadvantages of the combinatorial parallelization and divide-and-conquer ideas and explain why their combination attains more computational power. The improved parallel Nullspace Algorithm is used to compute up to nearly 50 million elementary flux modes for a metabolic network for yeast, a task which was previously not possible using either of the two approaches individually.