# A conversation at the Brilliant water cooler

Logic Level 3

Suyeon is thinking of 2 (not necessarily distinct) positive integers, $$x$$ and $$y$$, each of which is greater than 1. She tells Calvin the product $$P = xy$$ and Aaron the sum $$S = x + y.$$ The equally intelligent Calvin and Aaron then engage in a short discussion as follows:

• Calvin: "I cannot determine $$S$$ at this point."

• Aaron: "All right then, here's a hint; $$S$$ does not exceed $$20$$, and if that's all you need to know to uniquely determine $$S$$ then I will know what $$P$$ is."

• Calvin: "I am now able to uniquely determine $$S$$."

Find $$S + P$$.

###### This question was inspired by the long-suffering Calvin Lin (inside joke).
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