TITLE: n-ary version of gcd$(a,b)\space $lcm$(a,b)$ = $ab$
QUESTION [11 upvotes]: This question was motivated by pondering this lcm identity.
Consider that $\gcd(1,6,15) = 1$, but $\operatorname{lcm}(1,6,15)=30$, but $1\cdot 6\cdot 15 = 90$.  $(2,6,15)$ shows a similar phenomenon.
So what is the correct identity for $n$-ary $\gcd$/$\operatorname{lcm}$?

REPLY [9 votes]: Such generalizations of gcd & lcm identities occur frequently as problems. Below are a couple of examples - which include references to prior literature, e.g. to an inclusion-exclusion max/min proof in Polya & Szego's problem book. They're problem E2229 from the Monthly 77 1970 p. 402; problem 1344 from Math. Mag. 64 1990 p. 134.