Tony and Maria each chose a positive integer and told these to Lenny, but to nobody else. Lenny said: "Either the sum of your two numbers is 70 or the product of your two numbers is 70." (Because Tony and Maria have complete high school mathematics they quickly determine that is not possible for both the sum and product to equal 70.) Tony then said: "This is not enough information for me to know what Maria's number is." Next, Maria said: "I do not have enough information to know what Tony's number is." This information is surprisingly enough information to know what Maria's number is. What is Maria's number? Justify your answer.

9514 1404 393

  Maria's number is 35

Step-by-step explanation:

Tony will not be able to tell whether his number and Maria's add or multiply to give 70 if his number is a proper divisor of 70.

Once Maria determines that Tony's number is a proper divisor of 70, she can use her own number to determine if 70 is the sum or product, unless her number is both the difference of 70 and a divisor, and also a divisor of 70. The only such number is 35.

The scenario must play out as described if Maria's number is 35.

__

Proper divisors of 70 are ...

  {1, 2, 5, 7, 10, 14, 35}

The difference of 70 and these is ...

  {69, 68, 65, 63, 60, 56, 35}

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