A repeating decimal with a repeating cycle of three digits will convert to a rational number with a denominator of 999.
To explain why, consider a repeating decimal such as \(0.abcabcabc...\) where "abc" represents the three digits that repeat. When converting this repeating decimal to a fraction, the formula generally used is:
\[ \frac{N}{999} \]
where \(N\) is the integer formed by the repeating digits "abc". The reason for using 999 is derived from the fact that the denominator is \(10^n - 1\), where \(n\) is the length of the repeating part—in this case, 3. Since \(10^3\) is 1000, we subtract 1 to find the denominator:
\[ 10^3 - 1 = 1000 - 1 = 999 \]
Therefore, the correct answer is 999.