Product of ages is 72. Sum is the number on the building

We try the possible combinations of the product of ages and also the corresponding
sum.

72,1,1 - sum = 74
36,2,1 - sum = 39
18,2,2 - sum = 22
18,4,1 - sum = 23
9,8,1  - sum = 18
9,4,2  - sum = 15
8,3,3  - sum = 14
6,6,2  - sum = 14

Now here's the trick. The 2nd mathematician was NOT able to guess the ages
even after checking the sum. From the table above, you see that all the possible
combinations have unique sums except (9,4,2) and (8,3,3) which have the same
sum. So the door number must be 14 and the mathematician is still not able to
find which is the right combination.

3rd clue refers to "oldest" child which means that (6,6,2) is not possible - 
because there are 2 children with the same age of 6 and you can't refer to one
as the "oldest".

Hence the right combination is (8,3,3)

The ages are 8, 3 and 3.

:)