Solution: As in the hint, we show that the bus number must be larger than 12. Person A could be aged 48 with children of ages 1,3,4,4 or 2,2,2,6 or Person A could be 36 with children aged 1,1,6,6 or 1,2,2,9. Therefore the bus number cannot be 13. Adding children aged one to the lists above also proves that the bus number cannot be any larger than 13.

To prove that the bus number cannot be any smaller than 12 requires an exhaustive search. No number strictly less than 12 can be written c₁ + ... + c_k in two different ways such that the products c₁c₂...c_k are equal. Also the only two ways this can be done for 12 are 1,3,4,4 and 2,2,2,6 as given above. This is easily checked by computer, but if anyone has a better argument, please let me know.