When students asked their teacher how old her children​ were, she​ said, "I have three children. The product of their ages is 72 and the sum of their ages is the number of this​ room." The children then asked for the door to be opened to verify the room number. Then Carly told the teacher she needed more information to solve the problem. The teacher​ said, "My oldest child is good at​ math." Then Carly announced the correct ages of her​ teacher's children. What are the ages of the​ teacher's children?