This theorem guarantees the existence of (*) LCMs and GCDs and gives a simple way to find them.
This theorem can be proven by applying the Jordan-Holder theorem to Z-sub-n. Domains where an analogue of this theorem holds for ideals are named for Dedekind. An example where an analogous theorem to this one fails to hold is Z adjoined root -5. It can be proven by noting that all PIDs are UFDs and the integers are a PID. This theorem guarantees the existence of (*) LCMs and GCDs and gives a simple way to find them. A proof of the existence claim in this theorem first writes n as a nontrivial product if n is not prime, and then inductively factors that. For 10 points, name this theorem stating that every integer greater than 1 has a unique factorization into primes.
This theorem can be proven by applying the Jordan-Holder theorem to Z-sub-n.
fundamental theorem of arithmetic [prompt on any part of unique-prime factorization theorem; prompt on fundamental theorem]