Since $n_1,\dots,n_i$ are positive integers and $20$ is primarily decomposed as $20=2^2\cdot 5$, we quickly find the largest $n$, that is $5221111$.
But before finding the smallest, lets note since $20$ is a multiple of $5$ and no digits other than $0$ and $5$ aren't divisible by $5$. and since product of the digits is non-zero, one and only one of the digits is $5$, but by similar but not not same reasoning we can say either, one and only one digit is $4$, or, two and exactly two digits are $2$. in either ways the some of non-units is $9$. So, we have 4 other digits, each of which are unit. so the numbers we can write with the properties asserted, have either, 6, or 7 digits. of course the smallest among them should have 6 digit. and is $111145$