Word Problem, Help Please!

[ironkitten]

A principal decided to challenge two teachers. The principal thought of two lucky numbers between one and a hundred, added them and told the first teacher the sum, and then multiplied them and told the second teacher the product. Then the principal overheard this conversation;
Second teacher: " I'm stuck. I just can't figure it out."
First teacher: " I already knew you couldn't figure it out."
Second teacher: " Oh, is that so? If that is the case, then I know the answer."
First teacher: And now, so have I.

Based on the conversation, what are the lucky numbers?

I thought maybe I could start by looking at the product of two prime numbers.

 

[stapel]

This is an old exercise, and instruction and advice (in various permutations) is available online.

Eliz.

 

[ironkitten]

So this problem is just really the process of elimination. I need to make some type of systmatic list of numbers and see what works.