Pff i've lost a 2h on internet trying to find how do you compute the LCM of 2 fractions
For example LCM of 2/3 and 1/4 is 2 , while LCM of 3/2 and 1/4 is 3/2
Why i need this?
Because i need to find the period of this
Hmm, I've never seen the LCM defined for rationals like that. Anyway, you seem to have provided all the info necessary for a definition and its implications, and what I come up with is:
Suppose we have two positive rationals a'/b' and c'/d'. First reduce to lowest terms to get a/b and c/d. Then let e=lcm(b,d). Multiply both rationals by e to get integers f=ae/b and g=ce/d. The let h=lcm(f,g). Then the final answer is h/e.
I haven't tested this thoroughly or proven it, so if someone sees a problem with it, please say so. It "seems right" for the time being.
I think when you wrote you meant
My method is based on the idea that for positive integers a,b,k, we have . I haven't proven this but it seems true. Then extend it allowing a,b to be positive rationals, while keeping k a positive integer.
Edit: MathWorld says that is true; see Equation 19, the LCM is distributive. I suspect a proof would be pretty simple considering prime factorisations.
I think this can be proven to work because:
It is proven that for positive integers a and b, . Writing and reducing to lowest terms results in ; from this, it can be seen that taking either of the products you mentioned gives the lcm. Apparently this extends to the rational case.