Choose 5 primes arbitrarily: p1, p2, p3, p4, p5
Make five bins of size:
n1*p2*p3*p4*p5
n2*p1*p3*p4*p5
n3*p1*p2*p4*p5
n4*p1*p2*p3*p5
n5*p1*p2*p3*p4
where n1 < p1, n2 < p2, n3 < p3, etc...
We then make an arbitrary multiset consisting of one's, our five arbitrary primes, their pairwise products, and their three-way products (we don't have to have at least one of each element, and duplicates of elements are allowed).
If the total size of the elements in the multiset is less than (or equal to?) the total size of the five bins, is there always a way to pack these objects into the bins without overflowing?
What if we were to choose n primes, make n bins, and take numbers that are up to the product of (n-2) primes?