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Theorem \\ref{thm 2.3} is the main result which gives a necessary and sufficient condition for a partition set $\\{S_1,S_2,\\ldots, S_x\\}$ of $I_n$ with prescribed $|S_i|$ to be a constant sum partition of $I_n$, $1 \\leq i \\leq x$ and $n > x \\geq 2$. 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