Units digit of a random power

MediumProbability Combinatorics~6m

Pick $m$ uniformly from $\{111, 133, 155, 177, 199\}$ and, independently, $n$ uniformly from $\{2004, 2005, \dots, 2023\}$ . What is the probability that $m^{n}$ has units digit $1$ ?

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Only the last digit of $m$ matters. Find each last-digit's units-digit cycle.
Condition on the exponent's residue: $4\mid n$ for digits $3,7$ ; $n$ even for digit $9$ . Average over the five $m$ 's.

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Units digit of a random power

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