I Don't Know Your Number. Wait, Now I Do!

Logic Level 2

Alice thinks of two positive integers, secretly tells Bob one, and secretly tells Charlie the other. Then the following conversation occurs:

Alice: "The product of my numbers was 8. Or maybe it was 16."
Bob (to Charlie): "I don't know your number."
Charlie (to Bob): "I don't know your number."
Bob (to Charlie): "I don't know your number."
Charlie (to Bob): "I don't know your number."
Bob (to Charlie): "I know your number!"

What was Charlie's number?

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