Guessing a Two-Digit Number

Source: Sujayendu Patra (Blackrock Quant, ex-colleague at Morgan Stanley Strats and Modeling Team). Originally from the 26th PMO Qualifying Stage (Logic).

Problem:

Tasyo is thinking of a two digit number. He tells Basilio the tens digit and Crispin the ones digit, and challenges them to guess the number. Their conversation goes as follows:
- Crispin: I know the number is not prime.
- Tasyo: Correct, it is divisible by 3.
- Basilio: Now I know what the number is.

What digit did Tasyo tell Crispin?

Solution:

Highlight the part between the * symbols for the answer.
* The digit is 4. Crispin is sure the number is not prime, so every two-digit number ending in his digit must be composite. Endings 1, 3, 7 and 9 fail (11, 13, 17, 19 are prime), so his digit is one of 0, 2, 4, 5, 6, 8. The number is divisible by 3, so the tens digit has to fix the ones digit uniquely. If the tens digit is a multiple of 3, the ones digit could be 0 or 6. If the tens digit leaves remainder 1 (1, 4, 7), the ones digit could be 2, 5 or 8. If the tens digit leaves remainder 2 (2, 5, 8), only 4 works. Basilio knows the number, so the tens digit is 2, 5 or 8 and the ones digit is 4 (the number is 24, 54 or 84). (solution posted by AI) *

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