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Can I prove that the product of numbers like 7, 77, 777, ... cannot be a perfect square?

Original problem is the following but I assume this cannot happen.

You have a set of numbers of the form M = { 7, 77, 777, ..., 77....7 (n times) }.

Can it be that you have two disjoint sets A and B such as A + B = M have the property that the product of A's elements equals the product of B's elements?