Distinct Roots (not that kind)

Number Theory Level pending

Define \(M_i\) as the minimum positive integer such that all elements in the set \(S = \left\{M_i, \left\lfloor\sqrt[2]{M_i}\right\rfloor, ..., \left\lfloor\sqrt[i]{M_i}\right\rfloor\right\}\) are pairwise distinct. What are the last three digits of \(\sum _{i=2}^{10} M_i\)?

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