Competition problem — Number theory

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Number theory Difficulty 6.1 National olympiad Prove it

Example 4 Prove that 2221435693 cannot be divided by 9.

Translate the above text into English, keep the original text's line breaks and format, and output the translation result directly.

Solution

$\begin{aligned} 2221435693 & =2 \times 10^{9}+2 \times 10^{8}+2 \times 10^{7}+10^{5} \\ + & 4 \times 10^{5}+3 \times 10^{4}+5 \times 10^{3}+6 \times 10^{2} \\ +9 & \times 10+3 \end{aligned}$

By equations (13), (14) and Lemma 8, we have
$\begin{array}{l} 2221435693 \equiv 2+2+2+1+4+3+5+6 \\ +9+3(\bmod 9) \end{array}$

Since $2+2+2+1+4+3+5+6+9+3=37$ , which is not divisible by 9, we know from equation (15) that 2221435693 is not divisible by 9.

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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.