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Let $M = \{1, 2, \dots , n\}$. What would be necessary and sufficient condition(s) for the number $m$, so that $M$ can be expressed as the disjoint union of $m$ subsets $A_i$, $(i = 1, 2, \dots, m$), such that

(i) each $A_i$ contains the same number of elements, and

(ii) the sum of all elements of $A_i$ is the same for $i = 1, 2, \ldots, m.$

Obviously, $m\mid n$. Other than that, I don't know where to begin!

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    Well another rather trivial necessary condition is that $m\mid \frac{n(n+1)}{2}$2012-08-27

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