Digit Multiplication By K

There is a strange game played in ChefLand. The game starts with $N$ white balls, the $i$-th of which has a power of $S_i$. It is known that $0 \leq S_i \leq 9$. On each level, a black ball with power $K$ hits each of the white balls. After the collision, the power of each white ball is multiplied by $K$. However, white balls are only allowed to have single-digit power. In order to maintain th