To convert the repeating decimal \(0.123123123...\) into a fraction, we can denote it as \(x\):
\[ x = 0.123123123... \]
Notice that the decimal repeats every three digits. To eliminate the repeating part, we can multiply \(x\) by \(1000\):
\[ 1000x = 123.123123123... \]
Now, we have two equations:
- \(x = 0.123123123...\)
- \(1000x = 123.123123123...\)
Next, we can subtract the first equation from the second:
\[ 1000x - x = 123.123123123... - 0.123123123... \]
This simplifies to:
\[ 999x = 123 \]
Now, divide both sides by \(999\):
\[ x = \frac{123}{999} \]
We can simplify the fraction \( \frac{123}{999} \). The greatest common divisor (GCD) of \(123\) and \(999\) is \(3\):
\[ \frac{123 \div 3}{999 \div 3} = \frac{41}{333} \]
Therefore, the fraction that represents the repeating decimal \(0.123123123...\) is:
\[ \frac{41}{333} \]
None of the options provided other than a direct representation of \( x \) and the simplified format in fractional form, so there isn't a correct choice listed in your responses.
However, if we look for approximately or closest fractions based on available options you provided, none of them accurately represent the value of \(0.123123123...\) as \( \frac{41}{333} \).
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