Euler's Totient Rule

The number of bases mod $p$ in which $1/p$ has cycle length $l$ is the same as the number of Fractions $0/(p-1)$, $1/(p-1)$, ..., $(p-2)/(p-1)$ which have least Denominator $l$.

See also Totient Function

References

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 167-168, 1996.