66th Czech and Slovak Mathematical Olympiad

Number $2016$ is a multiple of $9$, therefore the digit sum of the resulting number has to be divisible by $9$. This happens if and only if we insert a number with digit sum divisible by $9$, that is a number which is a multiple of $9$. Trying out the smallest such numbers, we find out that inserting $9$, $18$, and $27$ doesn't work (instead of checking divisibility by $2016$ we can check that numbers $20916$, $201816$, and $202716$ are not divisible by $16$, looking at their final four digits). On the other hand, $203616=201600+2016$ is a multiple of $2016$. The answer is $36$.