Divisibility Tests and Recurring Decimals in Euclidean Domains

In this article, we try to explain and unify standard divisibility tests found in various books. We then look at recurring decimals, and list a few of their properties. We show how to compute the number of digits in the recurring part of any fraction. Most of these results are accompanied by a proof (along with the assumptions needed), that works in a Euclidean domain. We then ask some questions related to the results, and mention some similar questions that have been answered. In the final section (written jointly with P. Moree), some quantitative statements regarding the asymptotic behaviour of various sets of primes satisfying related properties, are considered. Part 1 : Divisibility Tests 1. The two divisibility tests: going forward and backward We are all familiar with divisibility tests for a few integers such as 3, 9, and 11. To test whether a number is divisible by 3 or 9, we look at the sum of its digits, which is equivalent to taking the weighted sum of the digits, the weights all being 1. For 11, the corresponding test is to examine the alternating sum and difference of digits, which means that the weights are 1,−1, 1,−1, . . . . The problem of finding a sequence of weights for various divisors has been dealt with in [Kh]; in this section, we briefly mention the various tests. First, some notation. Any s ∈ N with digits sj : 0 ≤ j ≤ m is a polynomial of 10, i.e. s = smsm−1 . . . s1s0 = m ∑