Let $G$ be an abelian group and let $H$ and $K$ be finite cyclic subgroups with $|H| = r$ and $|K| = s$. Show that $G$ contains a cyclic subgroup of order $lcm(r, s)$.
I have seen Let G be abelian, H and K subgroups of orders n, m. Then G has subgroup of order lcm(n,m). However, since it is required to show that $G$ contains a cyclic subgroup of order $lcm(r, s)$, I still have no clue.
This is an exercise problem on Fraleigh A First Course in Abstract Algebra (p.68, Exercise 56), after the section in which cyclic groups are introduced.