同学圈里有人出了道题，据说是美国小学四年级的数学题。结果在几个同学提出缺条件、和上网去查美国孩子几岁巧克力过敏未果之后，终于有个陪儿子读过奥数的同学出来帮我们做出来了，他说这也是上外附中小升初的口奥题目(口奥：奥数口算题)。

A travelling salesperson shows up at the apartment of a math teacher to peddle her wares.  During the course of her spiel, the salesperson discovers that the teacher has three children and inquires as to their ages. 

The teacher answers, "The sum of my children's age is 13."

The salesperson immediately declares, "That's not enough information to deduce your kids' ages!"

The teacher continues, "Well, the product of their ages is the same as my apartment number."

After taking a quick check of the apartment number, the salesperson thinks for a minute but then sighs, "That's STILL not enough information for me to figure it out."

So the teacher says, "Oh, I forgot to mention, my eldest child is allergic to chocolate."

As which point the salesperson exclaims, "Aha, now I know all their ages!"

What are the ages of the children?