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 1 of 5: The digit product never exceeds x
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In plain words

Adding extra digits after the leading one can only multiply the digit product by at most 99 each time, while it multiplies xx itself by roughly 1010; so for numbers with more than one digit, the digit product quickly falls far behind xx.

x=d1d2⋯dn‾  ⟹  d1d2⋯dn≤d1⋅10n−1≤xx=\overline{d_1d_2\cdots d_n} \;\Longrightarrow\; d_1d_2\cdots d_n \le d_1\cdot 10^{n-1} \le x
Detailed analysis

Write the decimal expansion of xx as d1d2⋯dnd_1d_2\cdots d_n, where d1≥1d_1\ge1. Since each digit is at most 9<109<10, the product d1d2⋯dn≤d1⋅10n−1d_1d_2\cdots d_n\le d_1\cdot10^{n-1}, and x≥d1⋅10n−1x\ge d_1\cdot10^{n-1} because d1d_1 is the leading digit. Hence the digit product is always at most xx, with equality only when xx has a single digit.