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.

Charlie

Thank you for introducing to the extensive world of essays triggered by the original puzzle.

I have solved this problem long time ago, read some article quoting (48,4) as the only possible ambiguous  set(age, #of children) corresponding to bus #12.

At that time I realized that there is something wrong about the uniqueness of the set (a,b,c),

 provided by you and other mathematicians i.e. (48, 12, 4).

I   wrongly remembered 36 as a possible age, and found support in Steve Herman's comment.

Did not check too much and left it to you. MEA CULPA, really sorry.

 However , the question regarding duplicity of ages  as created  by “propagation” , prevails.

What’s   wrong with (48, 13, 5) i.e. the guys are on bus number 13 and A,  aged 48 has 5 children?

I do not introduce quintuplets 1,1,1,1,1 with quadruplets 2,2,2,2  for a 16 yrs old father of 9 children riding on the same bus.

The ages are funny, but possible: 

13   48 5    1   1   3   4   4          

13   48 5    1   2   2   2   6   (from your listing) 

If I err, tell me where.

If not, how did the others get away with it?