Question 30
Let $n$ be a positive integer. When $\frac{n}{567}$ is expressed as a decimal, it is a recurring decimal with smallest period $k$. Find the sum of all possible values of $k$. (Note: We say that $k$ is a *period* of a recurring decimal if starting from some place the digits of the decimal repeat every $k$ digits. For instance, $0.12345345345\dots$ has smallest period 3 while $0.142857142587142587\dots$ has smallest period 6.)