The length of the repeating cycle of a repeating decimal is determined by the prime factorization of the denominator when the fraction is in its lowest terms. Specifically, the number of digits in the repeating cycle is dictated by the factors of 10 that the denominator does not include.
For a fraction \( \frac{a}{b} \) to have a repeating decimal, the denominator \( b \) must not only be composed of the primes 2 and 5. If \( b \) contains other prime factors, the decimal will repeat.
The maximum length \( d \) of the repeating cycle is given by the smallest positive integer \( d \) such that \( 10^d \equiv 1 \mod b' \), where \( b' \) is the part of \( b \) that is coprime to 10 (essentially, factors of \( b \) that are not 2 or 5).
To summarize, the length of the repeating cycle depends on the prime factors of the denominator and can range from 0 (for terminating decimals) to potentially any positive integer, depending specifically on the values that \( b \) can take and the computation involving its Euler's Totient function.
For example:
- For \( b = 3 \), the repeating cycle is of length 1 (0.333...).
- For \( b = 7 \), the repeating cycle is of length 6 (0.142857...).
- For \( b = 11 \), the repeating cycle is of length 2 (0.09...).
You can apply this principle to any denominator \( b \) to determine the potential length of the repeating decimal cycle.