Ignat gives two tutors a problem. He’s thinking of two different positive integers greater than \(1\), whose sum is at most \(100\). Ignat tells Pier (\(P\)) the product of the two numbers, and Sasha (\(S\)) the sum of the two numbers. Both Pier and Sasha know all of this, and then have a little conversation:
Sasha says that Pier doesn’t know the two numbers.
Pier says that he knows the two numbers.
Finally Sasha says that he knows the two numbers.
What are the two numbers?
On a bus out of the Hattius kingdom, you sit behind two wizards. The first says to the second “I’ve a positive integer number of children, all of whose ages are positive integers. The sum of their ages is the bus number, and their product is my age."
The second wizard replies “If you told me your age and how many children you had, would I be able to work out their individual ages?"
The first wizard says “No, unfortunately not." To which the second says “Aha! Now I know how old you are." What was the bus number?