## Tuesday, July 26, 2005

### Extended Midy's Theorem

Theorem (Original)

If the period of a repeating decimal for $a/p$, where p is prime and $a/p$ is a reduced fraction, has an even number of digits, then dividing the repeating portion into halves and adding gives a string of 9s. For example, $1/7$ = 0.142857…, and 142 + 857 = 999.

Theorem (Extended)

If the period of a repeating decimal for $a/p$, where p is prime and $a/p$ is a reduced fraction, is h = $m \times n$, then dividing the repeating portion into n parts and adding gives $c_n(a,p)$ × 9 m's, where c is a constant depending on p, a and n.