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In this paper we study the sequences defined by the last and the lastnon-zero digits of $n^n$ in base $b$. For the sequence given by the last digitsof $n^n$ in base $b$, we prove its periodicity using different techniques thanthose used by W. Sierpinski and R. Hampel. In the case of the sequence given bythe last non-zero digits of $n^n$ in base $b$ (which had been studied only for$b=10$) we show the non-periodicity of the sequence when $b$ is an odd primepower and when it is even and square-free. We also show that if $b=2^{2^s}$ thesequence is periodic and conjecture that this is the only such case.

On the last digit and the last non-zero digit of $n^n$ in base $b$