Extremal theorems on divisors of a number

AbstractFor every positive integer N, we determine the maximum sizes of collections C and C' of divisors of N subject to the following restrictions. I any two numbers in C are coprime, then their least common multiple must be N. If any k numbers (k arbitrary) in C′ have the greatest common divisor 1, then their least common multiple must be N. For the special case that N is square free, the maximum sizes of C and C' have previously been determined. Since every number can be thought of as a multiset of primes, this work can be regarded as an extension of theorems on families of finite sets to families of multisets