Some notes on divisibility rules

A divisibility rule is a shorthand way of determining whether a given number is divisible by a fixed divisor without performing the division, usually by performing a simple calculation on its digits. There is no similarity among these rules, in the sense that for example the rule for testing the divisibility by 3 is very different from the rule for 7. In this working paper we present a general rule for testing divisibility by two digit integer numbers. The general rule is characterized by a pair of conditions, but in some cases just the main condition of the two is necessary and sufficient for divisibility. In the second part we investigate on this aspect and we show in general what are the cases in which just one condition is enough. Although there are divisibility tests for numbers in any base, and they are usually different, in this paper we concentrate just on the base 10.