Solution

A complete factorization of the product of the ages (72) gives the following theoretical possibilities to a unique solution.
As Brian, who knew the street number and thus the sum, still could not tell the ages without further help, we can conclude that the street number must be 14 because this is the only sum that doesn't give a unique solution.
The sum is arrived at in two possible ways -- 2,6,6 and 3,3,8. When Brian says "The oldest ..." the combination 3,3,8 is only one possible solution for Brian and thus for us.
 
 
Youngest Middle Oldest
Sum
1
1
72
74
1
2
36
39
1
3
24
28
1
4
18
23
1
6
12
19
1
8
9
18
2
2
18
22
2
3
12
17
2
4
9
15
2
6
6
14
3
3
8
14
3
4
6
13

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Last updated: 2003-03 by Ingvar Jönsson