Let’s say a number $n$ is insertable if for every digit $d$, if we insert $d$ between any two digits of $n$, then the obtained number if a multiple of $d$.
For example, $144$ is not insertable because $1474$ is not divisible by $7$.
The question is the find the smallest insertable positive integer with at least two digits.
It is relatively easy to see that such a number have to be divisible by $2520$. I also ran a script to check all integers below 75,000,000,000 with no success.
Disclaimer. I do not know if such a number do exist.