Let $G = \langle e_1, \dots, e_n \rangle$ be a finitely generated abelian group. Denote by $n = \exp(G)$ the least common multiple of $\{\operatorname{ord}(e_1), \dots, \operatorname{ord}(e_n)\}$. Then $n$ is the LCM of the order of all elements of $G$.
How do you show that there is an element $a \in G$ of order n?
This should be done without using the fundamental theorem of finitely generated abelian groups, as I want to use this statement to prove a special case of the fundamental theorem.
For every prime power $p^k$ dividing $n$, we find $e_j$ such that $p^k\mid \operatorname{ord}(e_j)$, hence find a multiple with order $=p^k$. Verify that the sum of such elements for powers of different $p$ has order the product of the involved prime powers. If we take maximal prime powers, this product is the LCM.