Midy's Theorem

If the period of a Repeating Decimal for $a/p$ has an Even number of digits, the sum of the two halves is a string of 9s, where $p$ is Prime and $a/p$ is a Reduced Fraction.

See also Decimal Expansion, Repeating Decimal

References

Rademacher, H. and Toeplitz, O. The Enjoyment of Mathematics: Selections from Mathematics for the Amateur. Princeton, NJ: Princeton University Press, pp. 158-160, 1957.