Tell me something I don't know

Logic Level 3

Barry chose two not necessarily distinct integers from 1 to 6 inclusive.
He told Gerry only the sum of these two integers, and Harry only the product.
But Barry let both Gerry and Harry know that the numbers came from 1 to 6 inclusive.

The following conversation occurs:

Gerry: "I don't know the numbers."
Harry: "I don't know the numbers."
Gerry: "I don't know the numbers."
Harry: "I don't know the numbers."
Gerry: "Now I know the numbers!"

What is the product of these two numbers?


Inspiration.

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