To determine how many digits long a repeating cycle of a repeating decimal might be when the denominator of the rational number is 7, we need to consider the properties of the denominator relative to the base of the number system we are working in.
In this case, we are dealing with base 10 (the decimal system). The length of the repeating cycle for a rational number \( \frac{p}{q} \) can be determined by finding the smallest integer \( d \) for which \( 10^d \equiv 1 \mod q \), provided that \( q \) is coprime to the base (which in this case, it is because \( 10 \) and \( 7 \) are coprime).
For \( q = 7 \), we find \( d \) such that:
\[ 10^d \equiv 1 \mod 7 \]
We can calculate the powers of 10 modulo 7 to find \( d \):
- \( 10^1 \mod 7 = 3 \)
- \( 10^2 \mod 7 = 10 \mod 7 = 3 \)
- \( 10^3 \mod 7 = 30 \mod 7 = 2 \)
- \( 10^4 \mod 7 = 20 \mod 7 = 6 \)
- \( 10^5 \mod 7 = 60 \mod 7 = 4 \)
- \( 10^6 \mod 7 = 40 \mod 7 = 5 \)
- \( 10^7 \mod 7 = 50 \mod 7 = 1 \)
We find that \( 10^6 \equiv 1 \mod 7 \). This means the repeating decimal for any fraction with a denominator of 7 will have a repeating cycle of length 6.
Therefore, a repeating cycle of a repeating decimal for the denominator of 7 could be 6 digits long.
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