I was experimenting with an algorithm for generating random numbers from a discrete distribution and came across an interesting observation. Suppose that you have any finite set of rational numbers in simplest form that sums to one, such as
$\frac{1}{2} + \frac{1}{3} + \frac{1}{6}$
$\frac{1}{4} + \frac{1}{5} + \frac{11}{20}$
$\frac{1}{3} + \frac{1}{3} + \frac{1}{3}$
I noticed that in each case, there is some fraction in each sum whose denominator is the least common multiple of all the denominators collectively. In the first example, $6 = LCM(2, 3, 6)$; in the second, $20 = LCM(4, 5, 20)$; in the third, $3 = LCM(3, 3, 3)$.
In all of the examples I've tried, this pattern occurs, but I'm not sure if this is just a coincidence or not. I've attempted to prove that this is true, but I can't make much progress on it.
Am I completely wrong that one of the fractions must have a denominator that's the LCM of the denominators? Or is there a proof that might help shed light on why this is?
Thanks!
It's not always true: for example,
$\frac{1}{15}+\frac{1}{10}+\frac{5}{6}=1$
is a counterexample.