Last digit in $\sum_{k=1}^{999}k^m$ (olympiad question)

You can add $1000^m$ to the sum as it will not change the last digit as its last digit is $0$. Last digits of $1^m,11^m,21^m,...,991^m$ are the same. Similarly for $2^m,12^m,...,992^m$ and so on till $10^m, 100^m,...1000^m$. So the ones digit of $1^m+2^m+...+10^m$ is the same as that of $11^m+12^m+...+20^m$ and so on. There are $100~10$s in $1000$, so the ones digit of the sum is$$(1^m+2^m+...+10^m)*100\mod10$$which is $0$.