Modding the Digits

Call a number \(k\) AP'ED if it is of the form \(k=a^b\) with \(a\) and \(b\) integers such that \(a,b\ge2\), \(a\leq9,\) and the representation of \(k\) is \(\overline{N_1N_2N_3\ldots}\) where \(N_1\text{ mod } a\equiv N_2\text{ mod } a\equiv N_3\text{ mod } a\equiv\ldots\neq0.\)

Find the smallest AP'ED number.

I came up with this problem at the AP Calculus BC Exam today while playing around with my calculator, waiting for one of the sections to finish.
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