Perfect commons

Number Theory Level pending

\[\large \begin{cases} a = \sqrt[3]{\overline{Z \cdots X}} \\ b = \sqrt{\overline{Z \cdots X}} \end{cases} \]

Given that \(X\) is the last digit of a very large number, \(\overline{Z \cdots X}\) and \(a\) is a positive integer while \(b\) is not an integer.

If \(x_1,x_2,x_3,\ldots, x_n\), such that \(x_1<x_2<x_3< \ldots<x_n\), are the possible values of \(X\), submit your answer as \(\overline{x_1x_2x_3\ldots x_n}\).

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