1. ## Logic

Two integers m and n between 2 and 100 inclusive are chosen. The sum of the two integers is given to Sam, and the product of the two integers is given to Pam. Both Sam and Pam are brilliant mathematicians who make perfect logical deductions whenever there is a logical deduction to be made. After they look at their numbers for a while, Pam says "I don't know your sum, Sam." Sam responds "I knew you didn't know my sum, Pam." Pam thinks for a moment and then says, with a smile, "Now I know your sum." Sam's response is "Now I know your product." What are m and n?

2. Originally Posted by chiph588@
Two integers m and n between 2 and 100 inclusive are chosen. The sum of the two integers is given to Sam, and the product of the two integers is given to Pam. Both Sam and Pam are brilliant mathematicians who make perfect logical deductions whenever there is a logical deduction to be made. After they look at their numbers for a while, Pam says "I don't know your sum, Sam." Sam responds "I knew you didn't know my sum, Pam." Pam thinks for a moment and then says, with a smile, "Now I know your sum." Sam's response is "Now I know your product." What are m and n?

I found the solution to this thing, but I cannot offer a proof. Frankly, I didn't understand the explanations I found. Maybe someone more astute in Number Theory will tackle this one. The "unique" solution is below. Just highlight the area between the <> if you want to see!

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