# Maximum Length of Repeating Fraction Pattern

I have problem getting that maximum length of the repeating pattern of a fraction a to be constrained by c-1 if a=b/c

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#### Solution Preview

Firstly, we will state the following theorem:

Theorem

If p is a prime number other than 2 and 5, then the cycle length of 1/p is at most (p - 1), and the cycle length must divide (p - 1).

The problem

Now, let a be a rational number, and let b and c be integers such that b/c = a and c > 0 . Since multiplying a repeating decimal by a constant won't change that it's repeating, it's sufficient to show that 1/c has either a terminating or repeating decimal expansion.

Let c* be c with all factors of 2 and 5 removed, that is, c*∙ 2^m 5^n = c for some naturals m and n, and 2 and 5 do not divide c*. Then, 10 and c* are ...

#### Solution Summary

Finding maximum length of the repeating pattern of a fraction a to be constrained by c-1 if a=b/c