Generalizations of Midy’s Theorem on Repeating Decimals

Let n denote a positive integer relatively prime to 10. Let the period of 1/n be a · b with b > 1. Break the repeating block of a · b digits up into b sub blocks, each of length a, and let B(n, a, b) denote the sum of these b blocks. In 1836, E. Midy proved that if p is a prime greater than 5, and the period of 1/p is 2 · a, then B(p, a, 2) = 10 − 1. In 2004, B. Ginsberg [2] showed that if p is a prime greater than 5, and the period of 1/p is 3 · a, then B(p, a, 3) = 10 − 1. In 2005, A. Gupta and B. Sury [3] showed that if p is a prime greater than 5, and the period of 1/p is a ·b with b > 1, then B(p, a, b) = k · (10−1). (The results of Midy and Ginsberg follow quickly from this). In this paper we examine the case in which p is not necessarily prime. Define two positive integers u and v to be period compatible provided that there exist odd integers r and t and a positive integer s such that the periods of 1/u and 1/v are of the form r · 2 and t · 2 respectively. Let n be a positive integer relatively prime to 10 and let the period of 1/n be a · b with b > 1. The following are proved: (i) If n is relatively prime to 10 − 1, then B(n, a, b) = k · (10 − 1). (ii) If for every prime factor p of n, the integer a is not a multiple of the period of 1/p, then B(n, a, b) = k · (10 − 1). (iii) If b = 2, then B(n, a, 2) = 10 − 1 if, and only if, every two prime factors of n are period compatible.