wise men and math problem

Hello, dear people.

I want to offer you a rather interesting puzzle, I myself stumbled upon it quite a long time ago in an echo of a rug puzzle, and since then I have not met it either in my personal life or in the open spaces of the RuNet. Search on Habré also did not bring results.
Therefore, I offer it to your community in the form in which I myself read it for the first time.


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TWO SAGES

 A certain sultan had two wise men: Ali-ibn-Vali and Vali-ibn-Ali.
 Wanting to be convinced of their wisdom, the sultan called the wise men to him
 and said, "I’ve conceived two numbers. Both are whole, each more
 units, but less than a hundred.  I multiplied these numbers and the result will tell
 Ali and Vali while I say the sum of these numbers.
 If you really are as wise as they say about you, you can find out
 source numbers. "

 Wise men thought.  The first to break the silence of Ali.
 “I don’t know these numbers,” he said, lowering his head.
 “I knew that,” said Vali.
 “Then I know these numbers,” rejoiced Ali.
 - Then I know!  - Vali exclaimed.
 And the wise men informed the amazed king of the numbers that he intended.

 Identify them and you, oh reader.

From myself I want to add that it is quite possible to guess these numbers, but the proof of the uniqueness of the solution is a rather non-trivial task and there is a lot of routine and mathematics in it.

I would like to publish this puzzle in the “fun tasks” section, but I don’t have enough karma.

Oh yes, as they say here: this is my first post on Habré, for this I apologize for possible violations%)

upd: Attention! in comments already have an answer.
upd2: as suggested in comments, this task was on braingames.

upd3: well, the most complete answer that I know here