Choose 25 different positive integers no higher than fifty, such that none is a multiple of any of the others. What's the lowest total possible, and what's the set? (Note one such set would be 26 through 50 inclusive; however that set totals 950.)
The list I found turned out to be identical to the one Jer posted (total 711), though I started from the bottom up instead of high to low. I found one set allowing 25 entries, and then tried to improve it by adding a low number if it would knock out only one higher number (so there would still be 25 numbers, but the working total would be lower). This got 8, 12, and 14 onto the list, but no other numbers under 17 seemed to pay their way. Primes were obvious choices (since no lower factors), but 11 precluded 22 and 33, and 13 precluded 26 and 39. There may yet be a lower solution, but I think it will be similar to the list Jer posted. We'll see.
bottom up
