I'm afraid at this ungodly hour I've found myself a bit stumped. I'm attempting to answer the following homework question:
If $p_1,\dots,p_s$ are distinct primes, show that an abelian group of order $p_1p_2\cdots p_s$ must be cyclic.
Cauchy's theorem is the relevant theorem to the chapter that precedes this question...
So far (and quite trivially), I know the element in question has to be the product of the elements with orders $p_1,\dots, p_s$ respectively. I've also successfully shown that the order of this element must divide the product of the $p$'s. However, showing that the order is exactly this product (namely that the product also divides the orders) has proven a bit elusive. Any helpful clues/hints are more than welcome and very much appreciated!