Given $G \le C_{a_1} \times C_{a_2} \times \ldots \times C_{a_n}$. Is it true that there exists a generating set $S$ that generates $G$ has at most $n$ elements?
Indeed, the statement above is correct when $n=1$. Since $G\le C_{a_1}$ and $C_{a_1}$ is a cyclic group, $G$ is also cyclic. It follows that there exists $S=\{x\}$ generating $G$.