abstract algebra - On order of subgroups in abelian groups

I wonder whether any of you guys has already read the homonymous note by R. Beals in the December 2009 issue of the Monthly.

If so, would you be so kind as to let me know about the main ideas in Beal's approach? As you know, the whole point of his note is to present a solution to the following exercise in Herstein's Topics in Algebra:

Let G be an abelian group having subgroups of order m and n. Prove that G also possesses a subgroup of order lcm(m, n).

The funny thing about this proposal is that in subsequent editions of his book, Prof. Yitz would proclaim that he himself didn't have a solution using the authorized tools. Besides, he even went on to saying: "I've had more correspondence about this problem than about any other point in the whole book.".

Being aware of some of the history behind this little pearl, I'd really like to know what it is that Beals came up with. Is his approach crystal-clear? Is it somehow related to the standard attack of proving it first for the case gcd(m, n)=1?

Thanks in advance for you insightful replies.

P.S. The local library is the only access that I have to the literature. Unfortunately, they don't subscribe to any of the MAA periodicals.