Problem 2

Find all natural numbers

x

such that the product of their digits (in decimal notation) is equal to

x^2 - 10x - 22

Step 5 of 5: Conclusion: x = 12 is the unique solution

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In plain words

The whole argument is a template worth remembering: bound a discrete, slowly-growing quantity against a fast-growing algebraic expression to get a short finite list, then finish by direct verification instead of clever tricks.

\boxed{x=12} \text{ is the unique natural number with the required property}

Detailed analysis

Steps 1–2 show every solution must lie in $\{1,\dots,12\}$ , Step 3 eliminates $\{1,\dots,11\}$ , and Step 4 confirms $x=12$ works. Therefore $x=12$ is the one and only natural number whose digit product equals $x^2-10x-22$ .