Answer:
There are 7 solutions.
The possible answers are:
The oldest daughter is 72 and the others are 1 and 1 and the room number is 74.
The oldest daughter is 36 and the others are 1 and 2 and the room number is 39.
The oldest daughter is 24 and the others are 1 and 3 and the room number is 28.
The oldest daughter is 18 and the others are 1 and 4 and the room number is 23.
The oldest daughter is 12 and the others are 2 and 3 and the room number is 17.
The oldest daughter is 12 and the others are 1 and 6 and the room number is 19.
The oldest daughter is 9 and the others are 1 and 8 and the room number is 18.
The first few solutions are laughable and unlikely, but all are possible.
OR..
The clue that the sums of the daughters' ages is the room number is the sleeper. The student's first response is that two possible age combinations meet the necessary conditions. The teacher's reference to the "oldest daughter" resolves the ambiguity, so one of the solutions must consist of a set of ages in which at least two of the oldest daughters are the same age.
The prime factors of 72 are 2^3 *3^2. Because one of the solutions requires the two oldest daughters to be the same age, only combination 6, 6, 2 would be admissible. There are insufficient factors for any other combination in which the duplicate ages represent the oldest daughter. The ages of these daughters sum to 14, which is the room number. The other combination must therefore add to 14 and have an unique oldest age. The combination 8, 3, 3 meets all conditions and is the answer.
