A repeating decimal with a repeating cycle of three digits can be converted to a rational number with a denominator of 999.
To understand why, consider a repeating decimal like \( 0.\overline{abc} \) where \( abc \) represents the three digits that repeat.
When you express this repeating decimal as a fraction, it can be done by setting:
\[ x = 0.\overline{abc} \]
Multiplying both sides by \( 1000 \) (since there are three repeating digits), you have:
\[ 1000x = abc.\overline{abc} \]
Now, subtract the first equation from the second:
\[ 1000x - x = abc.\overline{abc} - 0.\overline{abc} \]
This simplifies to:
\[ 999x = abc \]
Thus, we have:
\[ x = \frac{abc}{999} \]
So, the denominator of the rational number representation of a repeating decimal with a repeating cycle of three digits is \( 999 \).
Therefore, the correct answer is A. 999.
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