Not sure if cycle is the best word here..(adsbygoogle = window.adsbygoogle || []).push({});

Let k and b (both > 1) be such that b does not divide k-1 (with the exception that b may equal k-1).

If n is the smallest power so that k^n is equivalent to 1 mod b (so n is the length of the digit cycle of 1/b in base k),

then bn is the smallest power so that k^(bn) is equivalent to 1 mod b^2 (so bn is the length of the digit cycle of 1/(b^2) in base k).

In base 10, 1/81 has length 9. 1/9 has length 1. I've checked many other examples.

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# Problem Dealing with Digit Cycles

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