### An Addition Theorem on the Cyclic Group ${\Bbb Z}_{p^\alpha q^\beta}$

#### Abstract

Let $n>1$ be a positive integer and $p$ be the smallest prime divisor of $n$. Let $S$ be a sequence of elements from ${\Bbb Z}_n={\Bbb Z}/n{\Bbb Z}$ of length $n+k$ where $k\geq {n\over p}-1$. If every element of ${\Bbb Z}_n$ appears in $S$ at most $k$ times, we prove that there must be a subsequence of $S$ of length $n$ whose sum is zero when $n$ has only two distinct prime divisors.