Last night I sat behind two wizards on a bus, and overheard the following:

A: I have a positive integral number of children, whose ages are positive integers, the sum of which is the number of this bus, while the product is my own age.
B: How interesting! Perhaps if you told me your age and the number of your children, I could work out their individual ages?
A: No.
B: Aha! AT LAST I know how old you are!

Rem: Taking in account the fatherhood limitations, this is uniquely solvable.

You are right, and I am relieved.

Now the only thing really wrong with the puzzle (and both of us let it go) is how fast A answered "No" to B's conditional questions.

Thanks again for removing unnecessary worries from my humble self.