divisibility by (n-1) in base n

I Just found out that if I want to check if a number in base $n$ is divisible by $n-1$, I just need to sum all the digits, again and again, until I get to a single character, and if this number is $n-1$, then this number is divisible by $n-1$.

For example, 45 in base 10 is divisible by 9, because the digits 4 + 5 = 9.

Why this happens?

I'm trying to prove it for base 16, and can't seem to get it right.