Digit Root and Remainders of Perfect Squares and Cubes

Let $n$ be any positive integer. Calculate the sum of its digits, then calculate the sum of the digits of that result, and continue this process until you obtain a single-digit number $R$.
a) Prove that $R$ is the remainder when $n$ is divided by 9.
b) What possible values can $R$ take if $n$ is a perfect square?
c) What possible values can $R$ take if $n$ is a perfect cube?