Competition problem — Number theory

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Number theory Difficulty 5.4 AIME, harder Prove it

Example 1. Prove that $(\sqrt{ } 26-5)$ "wears a decimal expression with at least $n$ consecutive teeth after the decimal point!":

Preserve the original text's line breaks and format, output the translation result directly.

Solution

$\begin{array}{l} \text { Prove that } 0 \leq \sqrt{ } 26-5=\frac{1}{\sqrt{ } 26}+5<\frac{1}{10}, \\ \therefore 0<(\sqrt{ } 26-5)^{\prime \prime}<\frac{1}{10^{D}}, \end{array}$

The conclusion holds.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.