Folklore

Let $P(x)$ denote the product of all digits of a natural number $x$, and $S(x)$ denote the sum of the digits of $x$.

How many solutions does the equation $P(P(x)) + P(S(x)) + S(P(x)) + S(S(x)) = 1984$ have?
Let $x$ be a natural number with the sum of digits $\underbrace{8 \ldots 80}_{248}$, containing several zeros in its notation. Then $P(x)=$ 0, hence $P(P(x))=S(P(x)))=0$. Moreover, $P(S(x))=0$, and $S(S(x))=1984$. Therefore, the equation has infinitely many solutions.

## Answer

Infinitely many solutions.