Your post just reminded me of something -

When writing a number in base 3, all numbers that are multiples of 2, when taking the sum of their digits in base 3, were even.

I tried this out with 4 and 3, and it worked there as well.

I then figured out a pattern that I forgot to ask about -

When writing a number in base x, do all numbers that are multiples of \(x-1\), when taking the sum of their digits, make another multiple of \(x-1\)?

This all started because of the trick that I used to figure out if a number was a multiple of 9 - sum up the digits until they are one. If it is 9, then it is a multiple of 9.