Jan 21, 2014 General Divisibility Criteria Anatoly A. Grinberg 1 and Serge Luryi Electrical and Computer Engineering Department University at Stony Brook, Stony Brook, NY 11794-2350 ABSTRACT We believe we have made progress in the age-old problem of divisibility rules for integers. Universal divisibility rule is introduced for any divisor in any base number system. The divisibility criterion is written down explicitly as a linear form in the digits of the test number. The universal criterion contains only two parameters that depend on the divisor and are easily calculated. These divisibility parameters are not unique for a given divisor but each two-parameter set yields a unique criterion. Well- known divisibility rules for exemplary divisors in the decimal system follow from the universal expression as special cases. I. Introduction Divisibility rules are designed to answer the question of the divisibility of a test integer A by a divisor n ― without actually performing the division. The usual rule corresponds to forming a criterion number C that is smaller than the test number (ideally, C << A) and possesses the same property in terms of the divisibility by n. Divisibility rules have been derived for many divisors [1, 2]. Some of these rules involve only a few rightmost digits of the test number. Thus the well-known criteria for divisibility into 2, 4, or 8 involve (in the decimal system) the 1, 2, or 3 rightmost digits, respectively. We shall be concerned with the rules involving all digits of the test number and presented as a linear form in these digits. For an (m+1)-digit test number A, ∑ = = m k k k a t A 0 , (I.1) where a k are its digits in the t-base number system, we seek the criterion of divisibility of A into n in the form ∑ = = m k k k a n c C 0 ) ( , (I.2) 1 Corresponding author: anatoly_gr@yahoo.com