Other Properties of Repetend Lengths
Various properties of repetend lengths (periods) are given by Mitchell and Dickson.
The period of 1/k for integer k is always ≤ k − 1.
If p is prime, the period of 1/p divides evenly into p − 1.
If k is composite, the period of 1/k is strictly less than k − 1.
The period of c/k, for c coprime to k, equals the period of 1/k.
If where n > 1 and n is not divisible by 2 or 5, then the length of the transient of 1/k is max(a, b), and the period equals r, where r is the smallest integer such that .
If p, p', p", … are distinct primes, then the period of 1/(pp'p"…) equals the lowest common multiple of the periods of 1/p, 1/p' ,1/p", ….
If k and k' have no common prime factors other than 2 and/or 5, then the period of equals the least common multiple of the periods of and .
For prime p, if but, then for we have .
If p is a proper prime ending in a 1 – that is, if the repetend of 1/p is a cyclic number of length p − 1 and p = 10h + 1 for some h – then each digit 0, 1, …, 9 appears in the repetend exactly h = (p − 1)/10 times.
For some other properties of repetends, see also.
Read more about this topic: Repeating Decimal
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