Please excuse the lack of expertise. I'm not a mathematician, nor have I studied it since high school.
I was thinking about how all the digits of multiples of $9$ summed equal a multiple of $9$.
I was musing on how miraculous this seemed, but I soon turned skeptical and wondered if it was a product of our base $10$ counting system.
I did a thought experiment, attempting to see if such a thing existed in a base $9$ counting system. It worked for the digit $8$. Subsequently, every time I checked in every counting system, the final digit ($x-1$, where $x$ is the base of the counting system) follows the same pattern.
For example, in a base $9$ system: $8 \times 2 = 17$
$1 + 7 = 8$
In a base $5$ system: $4 \times 3 = 22$
$2+2 = 4$
I'm wondering if this has a name. I tried looking it up, but I don't know enough mathematical jargon to figure out a decent search criteria.