What is the smallest multiple of $999$ with no $9$'s in it?
I can't find one by bashing, and I've just been plugging them into a calculator... Are there any smart ways?
Note: Please post a solution that isn't the answer or a bash. I already know the answer, but not how to get it. Thanks!
Let $X$ be the set of positive integer $n$ such that $999n$ doesn't contain any digit $9$. Let $m$ be the smallest element in $X$ and "$....abc$" be its decimal representation. It is clear $c \ne 0$. Otherwise, $\frac{m}{10} \in X$ contradicts with the choice that $m$ is the smallest element in $X$.
Let "$...def$" be the decimal representation of $999 m = 1000m - m$.
Since $c \ne 0$, it is easy to verify $$(d,e,f) = (9-a,9-b,10-c)$$ Since $999m$ doesn't contain any digit $9$, we have $$\begin{cases} d \ne 9 \implies a \ne 0 \implies a \ge 1\\ e \ne 9 \implies b \ne 0 \implies b \ge 1\\ f \ne 9 \implies c \ne 1 \implies c \ge 2 \end{cases} $$ So the smallest possible candidate for $m$ is $112$. By direct substitution, $999\times 112 = 111888$ doesn't contain any digit $9$. This mean $112$ is the smallest number we seek.