Two wise men and two numbers

Can you solve this beautiful puzzle?

A sultan had two sages, let's call them Saruman the White and Gandalf the Grey. Once the sultan decided to test how wise those sages are. He summoned them and said:

"I have secretly chosen two numbers. Both are integers greater than 1 but their sum is less than 100. The product of these numbers I'll tell to Gandalf, and the sum to Saruman. If you are really as wise as people say about you, you will know the numbers."

The sages began to think. The first to break the silence was Gandalf.
"I don't know the numbers," he said.
"I already knew that you didn't know," Saruman replied
"Oh, then I do know the numbers!" said Gandalf
"Then I also know them!" continued Saruman

And the sages told the amazed sultan his chosen numbers.

Determine the numbers.

The puzzle is known as Freudenthal Problem, Sum and Product Puzzle or Impossible Puzzle. Despite its name it is solvable. But before googling try to solve it yourself.

P. S. The picture is from "Lord of the Rings" movie.