## 轉貼美國卡內基美隆大學數學研究問題

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 $ab$ of the positive integers $a$ and $b$, and that the number given to Selma is the sum $a+b$ of the same numbers $a$ and $b$, where $b>a>1$. He doesn’t, however, tell Patty or Selma the numbers $a$ and $b$. The following (honest) conversation then takes place:

Patty: “I can’t tell what numbers $a$ and $b$ are.”

Selma: “I knew before that you couldn’t tell.”

Patty: “In that case, I now know what $a$ and $b$ are.”

Selma: “Now I also know what $a$ and $b$ are.”

Supposing that Homer tells you (but neither Patty nor Selma) that neither $a$ nor $b$ is greater than 20, find $a$ and $b$, and prove your answer can result in the conversation above.

Credit This problem comes from the Carnegie Mellon Math Studies Problem Seminar.

http://www.usamts.org/Solutions/Solution4_1_17.pdf

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