$X$ brags to her colleagues: "Both of my sons have their birthdays today. One of them is not yet 10 years old. Can you figure out how old they are? Anna, I'll whisper the product of their ages to you." (She whispers it.) Anna: "I still can't determine it from this." "Then I'll whisper the ratio of their ages to Bori." (She whispers it.) Bori: "I still can't determine it either." "You're right. I should have told Bori the age difference, then he could have figured it out."

How old are $X$'s sons?
Anna could not determine the ages of the children. This means that their product can be factored into the product of two natural numbers less than 10 in more than one way. As a first step, let's find all such pairs of numbers! We have listed those that give the same product under each other:

$$
\begin{array}{lllllllll}
1,4 & 1,6 & 1,8 & 1,9 & 2,6 & 2,8 & 2,9 & 3,8 & 4,9 \\
2,2 & 2,3 & 2,4 & 3,3 & 3,4 & 4,4 & 3,6 & 4,6 & 6,6
\end{array}
$$

Thus, Bori knew that the ages of the two children could only be one of the 9 pairs of numbers listed here. Mrs. X states that if Bori had been told the age difference, Bori could have figured out the ages of the children. Therefore, the age difference must appear only once among the differences derived from the 18 pairs of numbers. By examining the numbers, we find that only $4=6-2,6=8-2$, and the difference of $8=9-1$ appear once. Thus, Mrs. X's children are either 6 and 2, or 8 and 2, or 9 and 1 years old.

We also know that Mrs. X told Bori the ratio of the ages, and from this, Bori still could not determine the result. However, if the children were 6 and 2 years old or 9 and 1 years old, Bori could have selected the 6,2 or 9,1 solution from the (1) pairs based on the knowledge of the age ratio.

Thus, the children can only be 8 and 2 years old, which also satisfies the final condition of the problem: the ratio of the ages in the (1) pairs 1,4 and 2,8 is the same.