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The Josephus problem is concerned with anticipating which will be the last elements left in the ordered set \(\{1, 2, \ldots, n\}\) as successive $m$th elements (counting cyclically) are eliminated. We study the set of permutations of \(\{1, 2, \ldots, n\}\) which arise from the different orders of elimination as \(m\) varies, and give a criterion based on the Chinese Remainder Theorem for deciding if a given permutation can be interpreted as arising as a given order of elimination for some step size \(m\) in a Josephus problem.