the least amount
the smallest number

is how I would use the words

least number = smallest number

No difference, if the desired meaning is "the fewest"

On the other hand, if I write:

1, 2, 3, 4, 5

Here "1" is the smallest number on the list. You couldn't use "least" in this case.

I see dude's logic, but I find it mildly pedantic.

I think narbiepa has hit the nail on the thumb. (or on the head, as I suppose I should encourage foreigners to say.)