Answer: The bus line is 12 and the age of the first mathematician is 48.
Logical structure:
- The second mathematician knows the sum $S$ (bus number).
- If you also knew age $P$ and number of children $n$, you could list decompositions into positive integers with:
$$
a_1+\cdots+a_n=S,\quad a_1\cdots a_n=P.
$$
- The answer “No” means that, for the real pair $(P,n)$, there are at least two possible decompositions.
- After hearing that “No”, the second mathematician can deduce $P$; Therefore, with that $S$ there must be a single product compatible with that ambiguity.
Key case $S=12$:
With sum 12, the product that meets that property is:
$$
P=48.
$$
Because with $n=4$ there are two different options:
- $(1,3,4,4)$ with product 48,
- $(2,2,2,6)$ with product 48.
Thus, even knowing $(P,n)=(48,4)$, ages are not uniquely determined: exactly what the first mathematician claims.
For the other possible lines, this pattern does not allow the second to deduce a single age after hearing “No”.
The only line that makes the entire dialogue work is:
$$
S=12.
$$
That is why the second concludes correctly:
$$
\boxed{P=48}.
$$**Happy Idea:** The phrase “No” not only eliminates cases; encodes a condition of **structural ambiguity** on fixed-sum factorizations.
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$$