Let S be a finite set of positive integers which has the following property:if x is a member of S,then so are all positive divisors of x. A non-empty subset T of S is good if whenever x,y∈T and x<y, the ratio y/x is a power of a prime number. A non-empty subset T of S is bad if whenever x,y∈T and x<y, the ratio y/x is not a power of a prime number. A set of an element is considered both good and bad. Let k be the largest possible size of a good subset of S. Prove that k is also the smallest number of pairwise-disjoint bad subsets whose union is S. ratiofloor functionmodular arithmeticnumber theory proposednumber theory