Product makes Sum Parity Undecidable

Logic Level 3

Zach has chosen five numbers from the set {1,2,3,4,5,6,7}\{1, 2, 3, 4, 5, 6, 7\}.

If he told Claudia what the product of the chosen numbers was, that would not be enough information for Claudia to figure out whether the sum of the chosen numbers is even or odd.

What is the product of the chosen numbers?

Source: An old Olympiad Problem.

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