ILL 1977 Problem 11

Let $n$ and $z$ be integers greater than $1$ and $(n,z)=1$ . Prove: (a) At least one of the numbers $z_i=1+z+z^2+\cdots +z^i,\ i=0,1,\ldots ,n-1,$ is divisible by $n$ . (b) If $(z-1,n)=1$ , then at least one of the numbers $z_i$ is divisible by $n$ .