It is relatively easy to find a solution to the ages of Pavel's children and how long it has been since he saw Igor. Trial and error will do the trick. Proving that the solution is the only one that fits the narrative is much more difficult. Let's begin by reviewing the facts:

• Pavel has two children whose ages satisfy the relation $$(a + b) = n(a - b)$$ for whole numbers $$a, b,$$ and $$n$$.
• There ages also satisfy the relation $$ab = m(a + b)$$ for some whole number $$m$$.
• The older child, whose age is represented by $$a$$, is in high school.
• Some years ago, $$c$$, the children's ages at the time satisfied the relation $$(a - c + b - c) = p(a - c - (b - c))$$.

One thing that should be apparent is that if $$n = 1$$ or $$p = 1$$, then the younger child's age would have to be 0, i. e., either $$b = 0$$ or $$b = c$$. So $$n \ge 2$$ and $$p \ge 2$$.

We know from this page that $$a = {n + 1 \over n - 1}b$$, so for successive values of $$n$$ starting at $$n = 2$$, we can build a table of values for $$b$$ and $$a$$ and see which ones satisfy the other constraints of the problem. Once we reach a value for $$a$$ that exceeds high school age, we can move on to the next value for $$n$$.

 $$n = 2$$ $$b =$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$a =$$ $$3$$ $$6$$ $$9$$ $$12$$ $$15$$ $$18$$