56th Ukrainian National Mathematical Olympiad, Third Round

Obviously, all digits of such number are even because only even digit can be the last one. Also we cannot use $2$ or $6$ because if a number is divisible by $4$ then the last two digits of it are $12$, $32$, $...$, $92$ or $16$, $36$, $...$, $96$. So such number consists of the digits $4$ and $8$. So:

Three digits $4$ (or $8$) – two numbers.

Two $4$ and $8$ (or vice versa) – six numbers.

Overall – $8$ numbers.