Define a self dividing number in base n to be a number n digits long such that
- The digits $0-9$(depending on base) is used exactly once
- The first h digits are divisible by h,
for example in decimal we have
$3,816,547,290$
$(3/1=1),(38/2=19),(381/3=127),(3,816/4=954),...(3,816,547,290/10=381,654,729)$
other examples
$(10)_2,(1230)_4,(143250)_6$
Which is the least even base n that does not have this property?
In base $8$, there is $(32541670)_8$, so every even base up to $10$ is possible. A brute force search finds no solution in base $12$.
There is, by the way, a solution in base $14$: $$ (9\text{C}3\text{A}5476\text{B}812\text{D}0)_{14}, $$ where A,B,C,D represent the digits $10,11,12,13$, but none for bases $16,18,20$ or $22$.