Narcissistic numbers in other bases

Question

It is well known that $153$ is a narcissistic number; that is, it is equal to the sum of the cubes of its digits since $153=1^3+5^3+3^3$.
Other bases have similar numbers. For example, in base $3$, seventeen is $122$; and in base $4$, thirty-five is $203$.
Let $B_3$ be the set of bases with no such [edit] three-digit numbers. The first two members of $B_3$ are $2$ and $72$.
Why is every member of $B_3$ except $2$ a multiple of $9$?