Prove that an element has an order in an abelian group

2down votefavoriteI will quote a question from my textbook, to prevent misinterpretation:Let GG be a finite abelian group and let mm be the least common multiple of the orders of its elements. Prove that GG contains an element of order mm. I figured that, if |G|=n |G|=n, then I should interpret the part with the least common multiple as lcm (| x1 |, ...,| xn |) = m lcm (| x1 |, ...,| xn |) = m, where xi ∈G xi ∈G for 0≤i≤n 0≤i≤n, thus, for all such xi xi, ∃ ai ∈N ∃ ai ∈N such that m=| xi | ai m=| xi | ai. I guess I should use the fact that | xi | | xi | divides |G| |G|, so ∃k∈N ∃k∈N such that |G|=k| xi | |G|=k| xi | for all xi ∈G xi ∈G. I'm not really sure how to go from here, in particular how I should use the fact that GG is abelian.