Is it possible to color every positive real number so that

- each number is colored one of 10 distinct colors, and
- any two numbers that differ in exactly one digit are colored differently?

For example, the ten numbers \[{\color{maroon}0.111} \ldots,\, {\color{red}1.111}\ldots,\, {\color{orange}2.111} \ldots,\, {\color{limegreen}3.111} \ldots,\, {\color{green}4.111} \ldots,\, {\color{teal}5.111} \ldots,\, {\color{blue}6.111} \ldots,\, {\color{darkblue}7.111} \ldots,\, {\color{indigo}8.111} \ldots,\, {\color{magenta}9.111} \ldots\] must all be different colors because any two of them differ in exactly one digit. However, the numbers \(10.000 \ldots\) and \(1.000 \ldots\) do not have to be colored differently because they differ in more than one digit.

You may assume the axiom of choice holds.

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