Let $G $ be an abelian group of odd order, say $n$. Then, by fundamental theorem of finite abelian groups, $G$ is a direct product of groups of prime power order. Let $$ G=\mathbb Z_{{p_1}^{n_1}} \oplus \mathbb Z_{{p_2}^{n_2}}\oplus \cdots\oplus\mathbb Z_{{p_k}^{n_k}}.$$ Then $\vert G\vert=n={p_1}^{n_1}{p_2}^{n_2}\cdots{p_k}^{n_k}$. Now let $(g_1,g_2,...,g_k)\in G$. Then $\vert (g_1,g_2,...,g_k )\vert=\mathrm{lcm}(\vert g_1 \vert,\vert g_2 \vert,...,\vert g_k \vert)$.
From here how to move, i don't know but it is given in the hint to make use of fundamental theorem of cyclic groups (but i'm not getting how to apply it here).