Abstract
In this paper we deal with divisibility criteria for any integer in decimal system. In the development of these criteria we use facts from congruence theory: as modular Arithmetic, linear congruences, and some important properties of divisibility and congruence. Then, we give general divisibility criteria for the two classes of positive integers. The divisibility criteria for the first class of divisors is written down as a linear form in which the decades and the units digits of the test integer are involved in such a way that the co-efficient of the decades takes one and that of the units digit is an integer formed by a parameter, which is the solution of the linear congruence describing the co-primality of the divisor and the base of the underlying number system. This divisibility parameter is not unique, but each yields a unique criterion. Finally, we apply the rule giving a couple of examples and make a conclusion which summarizes the general divisibility test in terms of the two classes of divisors. Keywords: co-prime, modular Arthmetics, linear congruences, divisibility criteria, fundamental theorem of arthmetics DOI : 10.7176/JNSR/9-7-06 Publication date : April 30 th 2019
Highlights
In this paper we deal with divisibility criteria for any integer in decimal system
Are some facts of congruence theory,which is an important tool in number theory, besides handling related problems as solving congruence equations, remainder problems and the like, it is being used in the development of a generalized test of divisibility
The basic facts that are to be used in this paper are linear congruences and their properties along with modular Arthmetics and the Fundamental theorem of Arthmetics
Summary
If a and b are integers; the notation a≡ (modm) (“ais congruent to bmod m") means that a and b share the same remainder with respect to integer division by m, or, equivalently, that m|b−a. The condition m|b−a can be expressed as b=a+mq for some integer q.a is congruent to b mod m precisely if the difference of a and b is a multiple of m.An observation that will be useful later is that a≡(amodm)(modm).
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