Your solution still assumes we can break down the problem according to the fundamental theorem of arithmetic, which is a monoid homomorphism. You simply have ad

Your solution still assumes we can break down the problem according to the fundamental theorem of arithmetic, which is a monoid homomorphism. You simply have added more insight into the problem than I. The goal of the article is show there's a consistent structural framework of problem solving underneath. I mention briefly that you can improve the solution along your lines. The idea that let's us prove your algorithm above correct using this theory is that monoid morphisms compose (they form a functor category), and that equivalence relations that respect monoid composition induces monoid morphisms to the quotient classes. Then we just define the equivalnce relation where two numbers are equal if they have the same number of fives in their prime factorization. The composite monoid morphism yields your algorithm. I didn't want to overcomplicate the article with this, but I did mention it (it's in the hard-to-parse paragraph at the end of the section)