Yes.

A number in base $n$ is just $a*n^2+b*n^1+c$.

Because $n mod n-1 = 1$, you can replace every $n$ with $1$.

So $a*n^2+b*n^1+c mod n-1$ becomes $a+b+c mod n-1$

The trick for 11 generalizes for testing for divisibility by n+1 in base n, too

yes, n^k-1 is always divisible by n-1.