Note: This question was inspired by this projectEuler problem, but I am not looking for a way to solve it.
Why is it that the last $n$ non-zero digits do not repeat themselves in a simple fashion. After all, lets say I am looking for the last non-zero digit. I will always be multiplying by the same set of numbers in the same order, (01-99) and so I would assume that since all of these numbers will be hit once for 100!, twice for 200!, three times for 300! and four times for 400!, the last digit of these numbers would be scalar multiples of each other (mod 10) but this is not the case.
I sense that I am making a simple logical error. What is it?