Units Digit of a Power

Only the units digit of $m$ matters, so we may replace the set with $\{1,3,5,7,9\}$. Each of the five values is equally likely.

The set $\{2004,\dots,2023\}$ contains $20$ consecutive integers. Since $2004$ is divisible by $4$, each residue of $n$ modulo $4$ occurs exactly $5$ times. So $n \bmod 4$ is uniform on $\{0,1,2,3\}$ and independent of $m$.

Now look at the units digits of successive powers of each digit:

• $1$: always $1$. Then $m^n$ ends in $1$ for every $n$, with probability $1$.
• $3$: the cycle is $3,9,7,1$. The units digit is $1$ exactly when $4\mid n$, with probability $\tfrac14$.
• $5$: always $5$, with probability $0$.
• $7$: the cycle is $7,9,3,1$. The units digit is $1$ exactly when $4\mid n$, with probability $\tfrac14$.
• $9$: the cycle is $9,1$. The units digit is $1$ exactly when $n$ is even, with probability $\tfrac12$.

Averaging over the five equally likely values of $m$ gives

Key idea. Units digits of powers repeat with period dividing $4$. Because the range of $n$ covers whole periods, only the residue $n \bmod 4$ matters, and it is uniform.