22. (CZS 3) ${ }^{\mathrm{IMO} 2}$ Find all positive integers $x$ for which $p(x)=x^{2}-10 x-22$, where $p(x)$ denotes the product of the digits of $x$.
22. It can be shown by induction on the number of digits of $x$ that $p(x) \leq x$ for all $x \in \mathbb{N}$. It follows that $x^{2}-10 x-22 \leq x$, which implies $x \leq 12$.  Since $0<x^{2}-10 x-22=(x-12)(x+2)+2$, one easily obtains $x \geq 12$. Now one can directly check that $x=12$ is indeed a solution, and thus the only one.