The length of the repeating decimal for a rational number depends on the denominator when the fraction is in its simplest form. For a denominator of 7, the repeating decimal can have a maximum period of \( n \), where \( n \) is the smallest integer such that \( 10^n \equiv 1 \mod 7 \).
Calculating this:
- \( 10^1 \equiv 3 \mod 7 \)
- \( 10^2 \equiv 2 \mod 7 \)
- \( 10^3 \equiv 6 \mod 7 \)
- \( 10^4 \equiv 4 \mod 7 \)
- \( 10^5 \equiv 5 \mod 7 \)
- \( 10^6 \equiv 1 \mod 7 \)
Since \( n = 6 \) is the smallest integer where the equivalence holds, the repeating decimal can be up to 6 digits long.
Thus, the correct answer is:
A. Up to six digits long.
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