A conversation at the Brilliant water cooler
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 \(S + P\).