Question 43
The number $0$ is written on a blackboard. Every minute, Aerith simultaneously replaces every $0$ with a $1$ and every $1$ with a $10$. For example, if the current number were $1100$, in one minute it would be $101011$. She eventually gets tired and leaves, leaving some number $N$ written on the board. If $9 \mid N$, show that $99 \mid N$.