Two mathematicians, a visible route number, and a single denial that reveals one passenger's age.
The Conway Bus
Riddle statement
Two mathematicians are riding a bus, and both can see the route number.
One of them says:
—I have at least two children. Their ages are positive integers and may be repeated. The sum of their ages is this route number; their product is my age.
The other replies:
—If I also knew your age and how many children you have, I could determine their ages exactly.
—No, you could not.
After thinking for a moment, the second mathematician concludes:
—Then I know how old you are.
What is the route number, and what age has been deduced?