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 4 of 5: x = 12 satisfies the equation exactly

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In plain words

It is worth pausing here: with only twelve candidates in play and eleven already eliminated on sign alone, this final direct computation is a two-line arithmetic check, not a search.

x=12:\quad 12^2-10\cdot12-22=2, \qquad 1\cdot 2=2

Detailed analysis

For $x=12$ , the right-hand side is $12^2-10\cdot12-22=144-120-22=2$ . The digits of $12$ are $1$ and $2$ , whose product is $1\cdot2=2$ . The two sides agree, so $x=12$ is a genuine solution.