Talk:Product of GCD and LCM/Proof 2

Use of LCM of coprime m,n

The proof appears to use LCM equals Product iff Coprime for lcm(m,n) = mn which would make the proof circular atm.

(I believe) it can be adjusted by using Product of Coprime Factors (which comes from Bezout's Lemma), mn divides k, so gmn is the least common multiple. Qwr (talk) 16:55, 8 September 2021 (UTC)

Well spotted. Please feel free to amend it as necessary. --prime mover (talk) 18:05, 8 September 2021 (UTC)

I think the proof as presented with the existence of gk is a little confusing (rereading it makes it make more sense). What I think is clearer is something like: let x be a common multiple of a and b. Since a = gm and b = gn, x is divisible by g so can be written gk. Then after establishing m,n divides k, Product of Coprime Factors shows mn | k, so gmn is the least common multiple. I prefer to use more words than symbols but that is a stylistic choice. Qwr (talk) 18:44, 8 September 2021 (UTC)