Homer gives mathematicians Patty and Selma each a different integer, not known to the other or to you. Homer tells them, within each other’s hearing, that the number given to Patty is the product of the positive integers and , and that the number given to Selma is the sum of the same numbers and , where . He doesn’t, however, tell Patty or Selma the numbers and . The following (honest) conversation then takes place:
Patty: “I can’t tell what numbers and are.”
Selma: “I knew before that you couldn’t tell.”
Patty: “In that case, I now know what and are.”
Selma: “Now I also know what and are.”
Supposing that Homer tells you (but neither Patty nor Selma) that neither nor is greater than 20, find and , and prove your answer can result in the conversation above.
Credit This problem comes from the Carnegie Mellon Math Studies Problem Seminar.