4. Let $P(n)$ and $S(n)$ denote the product and sum of the digits of the natural number $\mathbf{n}$, respectively. For example, $P(133)=9, S(133)=7$. Find all two-digit numbers $\boldsymbol{n}$ for which the equality $n=P(n)+S(n)$ holds.
Solution. Let the desired two-digit number be $n=\overline{a b}=10 a+b, a \neq 0$. The equation from the condition takes the following form:

$$
10 a+b=a b+a+b \Leftrightarrow 10 a=a(b+1) \Leftrightarrow 10=b+1 \Leftrightarrow b=9
$$

Therefore, the condition is satisfied by all two-digit numbers ending in 9, and only by them.

Answer: $19,29,39,49,59,69,79,89,99$.