Midy's theorem

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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 A028416 in the OEIS). If the period of the decimal representation of a/p is 2n, so that a p = 0. a 1 a 2 a 3 … a n a n + 1 … a 2 n ¯ {\displaystyle {\frac {a}{p}}=0.{\overline {a_{1}a_{2}a_{3}\dots a_{n}a_{n+1}\dots a_{2n}}}} 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, a i + a i + n = 9 {\displaystyle a_{i}+a_{i+n}=9} a 1 … a n + a n + 1 … a 2 n = 10 n − 1. {\displaystyle a_{1}\dots a_{n}+a_{n+1}\dots a_{2n}=10^{n}-1.} For example, 1 13 = 0. 076923 ¯ and 076 + 923 = 999. {\displaystyle {\frac {1}{13}}=0.{\overline {076923}}{\text{ and }}076+923=999.} 1 17 = 0. 0588235294117647 ¯ and 05882352 + 94117647 = 99999999. {\displaystyle {\frac {1}{17}}=0.{\overline {0588235294117647}}{\text{ and }}05882352+94117647=99999999.}