Hexadecimal magic fraction

In decimal notation, \[\dfrac 1{81} = 0.01234567901234567901234\dots\]

In analogy to this, we wish to find a fraction for which in hexadecimal notation \[\dfrac 1{n} = 0.0123456789ABCDF0123456789ABCDF01\dots\] What is the value of \(n\), expressed as a hexadecimal numeral?

Bonus: Generalize this for number base \(b\).

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