divisible by 9?

Last week I posted a question about a trick I found to factoring a number ending in 5. Why did this trick work? The mathematicians of Reddit immediately had the answer:

Realize that a number ending in 5 can be written 10n+5. So

5q = 10n + 5,

q=2n+1 , where q is the desired factor

Brilliant minds out there. How beautiful.

Here is another question about factoring: I was told as a child that if a number has 9 as a factor, then the sum of the digits comprising the number will add up to a mulltiple of 9.

Example: 972

9+7+2 = 18, a multiple of 9

Is there some way to prove this?