Numbers In A Room

Guess its time for a new puzzle, so I can claim to post at least once a month. This is one Ben showed me, but I am rewording it heavily from the original:

Walking through a building, you overhear a conversation two people are having in a room:
Person 1: I have written down a list of positive integers whose sum is the room number of this room.
Person 2: If you were to tell me how many numbers were on the list and their total product, would I be able to figure out what numbers were on the list, given that I already know the room number we are in?
Person 1: No, you would not.
Person 2: Alright, I now know the total product of the numbers.
What is the smallest room number they could be in?

The "theme" might be a bit thin, but it wasn't any better in the original, and at least this makes it clearer.