Midy's theorem
In mathematics, Midy's theorem, named after French mathematician E. Midy, is a statement about the decimal expansion of fractions a/p where p is a prime and a/p has a repeating decimal expansion with an even period (sequence in the OEIS). If the period of the decimal representation of a/p is 2n, so that
then the digits in the second half of the repeating decimal period are the 9s complement of the corresponding digits in its first half. In other words,
For example,
If k is any divisor of the period of the decimal expansion of a/p (where p is again a prime), then Midy's theorem can be generalised as follows. The extended Midy's theorem states that if the repeating portion of the decimal expansion of a/p is divided into k-digit numbers, then their sum is a multiple of 10k − 1.
For example,
has a period of 18. Dividing the repeating portion into 6-digit numbers and summing them gives
Similarly, dividing the repeating portion into 3-digit numbers and summing them gives
Midy's theorem and its extension do not depend on special properties of the decimal expansion, but work equally well in any base b, provided we replace 10k − 1 with bk − 1 and carry out addition in base b.
For example, in octal
In duodecimal (using inverted two and three for ten and eleven, respectively)
Short proofs of Midy's theorem can be given using results from group theory. However, it is also possible to prove Midy's theorem using elementary algebra and modular arithmetic:
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