HMMT November 2024 E6

A positive integer $n$ is stacked if $2 n$ has the same number of digits as $n$ and the digits of $2 n$ are multiples of the corresponding digits of $n$. For example, 1203 is stacked because $2 \times 1203=2406$, and $2,4,0,6$ are multiples of $1,2,0,3$, respectively. Compute the number of stacked integers less than 1000 .