MathLabs

Problem 2

Find all natural numbers xx such that the product of their digits (in decimal notation) is equal to x2−10x−22x^2 - 10x - 22.
Step 3 of 5: Ruling out x = 1 through 11
Previous stepNext step
In plain words

This step is a cheap sign check, not an arithmetic search: there is no need to compute any digit products at all here, since negativity alone already disqualifies every one of these eleven candidates at once.

x∈{1,…,11}  ⟹  x2−10x−22<0, impossible for a digit productx\in\{1,\dots,11\} \;\Longrightarrow\; x^2-10x-22<0,\ \text{impossible for a digit product}
Detailed analysis

A direct check shows x2−10x−22<0x^2-10x-22<0 for every x∈{1,2,…,11}x\in\{1,2,\dots,11\} (for instance at x=11x=11: 121−110−22=−11121-110-22=-11, and the expression is even more negative for smaller xx near the vertex of the parabola). But a product of digits is a product of nonnegative integers, so it can never be negative. Hence none of x=1,…,11x=1,\dots,11 can satisfy the equation.