When is the lcm of a fraction sum the actual denominator.
Consider a sum
where each fraction is reduced. Alternatively using the familiar process of lowest common denominators, we have
where denotes the gcd and denotes the lcm. My question is, when is it true that ? For example, this does not hold for
Are there any simple necessary and sufficient conditions for ?
Edit It has been suggested that
is a necessary and sufficient condition. I'm interested in either a proof or a counterexample if possible.
Edit 2 After a bit of searching, I've found
as a counterexample.
I am still looking for nice conditions for this to be satisfied and I feel that I should give an explanation of exactly what type of condition I am seeking. Angela has provided a necessary and sufficient condition, but it does not seem to be "simpler" than simply adding the fraction and seeing if it reduces. This is perhaps ambitious, but I am looking for a condition which is simple enough to use by inspection for simple fractions.