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: "O.k., based solely on the fact that \(S\) does not exceed \(20\) 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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