Congruence and Divisibility: Divisibility Criteria for Positive Integers

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. Key words: co-prime, modular Arthmetics, linear congruences, divisibility criteria, fundamental theorem of arthmetics DOI : 10.7176/MTM/9-6-01 Publication date : June 30 th 2019


Introduction
Divisibility rules are designed to answer the question of divisibility of an integer a by a divisor integer without actually performing division. There are lots works that had been done in the field of number theory. But in the area of divisibility very little attention has been given. Although, for checking that a given integer is a multiple of any other integer is still time taking, we have some algorithms, such as Euclid's algorithm, which is one of the preeminent methods ever known regarding the underlying concept. Till the date there was no a feasible generalized test for divisibility. Here 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. In section four ,we show an application for the main result. A conclusion is given in the last section of this paper 2. Congruence and its properties, and basic notions 2.1. Congruence Definition2.1. If and are integers; the notation ≡ (mod ) (" is congruent to mod ") means that and share the same remainder with respect to integer division by , or, equivalently, that | − . Definition 2.2. Let , , be integers with > 0, then we say is congruent to modulo m iff | − . Symbolically, ≡ ( ).
Examples 2.1. Congruence between two integers a) 3≡5mod2, b) 23≡37 (mod7). Remark 2.1. Here, we see why the above two definitions are equivalent. If and have the same remainder (mod ), then = + for some integer and some0 ≤ < , and = + for some integer and the same . Therefore, − = ( − ) , which means that | − . Conversely, if | − , then − = for some integer .Then let = .It follows that = + for some integer . But then, = + = + + = + + Since is the remainder of (mod ), 0 ≤ < , and therefore, since = + 1 + , is also the remainder of (mod ). The condition | − can also be expressed as = + for some integer .Therefore, is congruent to mod precisely if the difference of and is a multiple of .An observation that will be useful later is that ≡( mod )(mod ). This follows directly from the definition.

Properties of Congruence
We now study how congruence interacts with the arithmetic operations of addition and multiplication. Theorem 2.1. if ≡ ( ) and ≡ ( ), then i.
ii. ≡ ( ) Proof: As we just stated on the previous remark, the assumptions of the theorem are equivalent to = + for some integer and = + for some integer . By Then it follows that + = + + + = + + ( + ) . That proves the first conclusion of the theorem. Similarly, we get = + + + . That proves the second conclusion of the theorem. It is a consequence of this theorem that in any computation of a remainder of some additive and/or multiplicative combination of integers, the integers involved can be reduced to remainders first. We will explain this later. Definition 2.3. We define relation ′ ≡ ′ as ≡ ( a. If a + k ≡ b + k (mod n) for any integer k, then a ≡ b (mod n) b. If k a ≡ k b (mod n) and k is coprime with n, then a ≡ b (mod n) Definition 2.4. The modular multiplicative inverse is defined by the following rules: There exists an integer denotedsuch that a -≡ 1 (mod n) if and only if a is coprime with n. This Integeris called a modular multiplicative inverse of a modulo n. o If a ≡ b (mod n) andexists, then -≡ -(mod n) (compatibility with multiplicative inverse, and, if a = b, uniqueness modulo n). In particular, if p is a prime number then is coprime with p for every such that 0 < < . Thus, a multiplicative inverse exists for all that are not congruent to zero modulo p.

Linear congruence
Definition 2.5. The congruence of the form ! ≡ ( ) is called a linear congruence with one variable x.
Definition 2.6. By a solution of the linear congruence ! ≡ ( Example 2.3. The solution of the linear congruence in the above example is 3. Theorem 2.4. The linear congruence ! ≡ ( ) has solution if and only if ' ( , )| .

Remark 2.2.
If a x ≡ b (mod n) and a is coprime to , the solution to this linear congruence is given by Example 2.4. Solve a linear congruence: find a solution to 8x ≡ 1 (mod 11). If there is an answer, it can be represented by one of 0, 1, 2, …. , 10, so we can just run through the possibilities: x mod 11 0 1 2 3 4 5 6 7 8 9 10 8x mod 11 0 8 5 2 10 7 4 1 9 6 3 The only solution is 7 mod 11: 8 × 7 = 56 ≡ 1 11. This means 7 and 8 are multiplicative inverses in . This problem concerns finding an inverse for 8 modulo 11. We can find multiplicative inverses for every nonzero element of : x 1 2 3 4 5 6 7 8 9 10 ! ( 1 6 4 3 9 2 8 7 5 10 Check in each case that the product of the numbers in each column is 1 in .  Vol.9, No.7, 2019 Example 2.5. Find a solution to 8! ≡ 1 ( 10). We run through the standard representatives for & ), and find no answer: x mod 10 0 1 2 3 4 5 6 7 8 9 10 8x mod 10 0 8 6 4 2 0 8 6 4 2 0 In retrospect, we can see a priori why there shouldn't be an answer. If 8! ≡ 1 10. for some integer x, then we can lift the congruence up to Z in the form 8x + 10y = 1 for some , ∈ . But this is absurd: 8x and 10y are even, so the left side is a multiple of 2 but the right side is not. Example 2.6. The linear congruence 6! + 1 ≡ 4 ( 15) has three solutions! In the following doesn't have to behave like real linear equations: there may be no solutions or more than one solution. In particular, taking b = 1, we can't always find a multiplicative inverse for each nonzero element of . . The obstruction to inverting 8 in & can be extended to other cases in the following way. Theorem 2.4. For integers a and m, the following are equivalent: i.
There is a solution x in Z to ! ≡ , ii.

Modular Arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value called the modulus. The modern approach to modular arithmetic was developed by Carl Frederich Gauss in his book Disquisitiones Arithmeticae in 1801.
In some applications, we are only interested in the remainder of some arithmetic operation. A familiar use of modular arithmetic is in the 12 hours clock, in which the day is divided in two 12 hours periods. For instance, if the time is 10:00 now, then after 5 hours it will be 3:00. Usual addition would suggest that the later time should be 10+5=15 but this is not the case because clock time "wrap around" every 12 hours. Because the hour number starts over after it reaches 12, this is arithmetic modulo 12. According to definition of congruence, 12 is congruent not only to itself, but also to 0, so the time is called "12:00" could also be called "0:00", since 12 ≡ 0( 12).
As it is easy to verify, the proof is left to the reader.
We took what appears to be a detour through equivalence relations because those three properties allow us to define addition, subtraction, and multiplication for congruences. Addition, subtraction, and multiplication work exactly the same way as they do with integers with the only constraint being that addition, subtraction, and multiplication is only allowed when the congruences have the same moduli. Example 2.8. Consider 14≡ 4(mod 10) .Here, we cannot divide both sides by two because 7 ≢ 2( 10). In other words, 14≡ 4(mod 10) fails to divide by 2 because both 2 and 10 are divisible by 2. Again, we can only divide provided that there are no common divisors between the number we are trying to divide by and the modulus. Note that if the modulus is a prime number then division is defined for all divisors.

Fundamental theorem of Arithmetic
Every natural number is built, in a unique way, out of prime numbers. Note that primes are the products with only one factor and 1 is the empty product. Theorem 2.5. Every natural number can be written as a product of primes uniquely up to order. Proof: An interested reader can establish the proof of this theorem using Mathematical induction and for the uniqueness part, also using proof by contradiction. Example 2.9. Write the natural number n=2775 as a product of distinct primes. The prime factorization of 2775 = 3 × 5 × 37.

Main Result
As far as our concern that we are developing a test for divisibility of integers co-prime to 10 Proposition 3.1. Let be a positive integer co-prime to 10, then there is an integer x such that 10! ≡ 1( ). (1) Proof: As ' ( , 10) = 1, then by GCD 1 -Theorem there are integers x and y such that 10! + , = 1. But in view of congruence theory, we obtain that 10! ≡ 1( ).
Thus, our main task here is finding such an integer x satisfying the congruence equation in the above proposition.
In performing this task of developing the criteria we require to solve the linear congruence using cancellation law in congruences. Clearly, it has solution because it satisfies the existence theorem for solution of linear congruences. ≡ 0( ) ⟹ A ≡ 0( ). ∎ Remark 3.2. According to the fact (i) in Remark 3.1., the divisibility criterion for m is not unique. For instance, for m=9, besides what is given under special divisibility criteria for integers co-prime to 10, we have at least one criterion b-17 & ≡ 0( 9).

Special divisibility criteria for integers coprime to 10
From the generalized divisibility criteria we extracted the special ones for few positive integers discussed as follows: Let A be a test number and m be a composite positive integer co prime to 10. Suppose, 9 = 10 7 7 + 10 7( 7( + ⋯ + 10 + 10 + & .Let b= 7 7( … (i.e. decades) and & is units digit. Then 9 = 10 + & . Here we give Divisibility criteria for 9, 21, 27 and 33 as follows

Special divisibility criteria for integers not coprime to 10
In this subsection we discuss divisibility criteria for those positive integers not relatively prime to 10. One may ask for what these integers are. Obviously, they are those integers which are multiples of 2 and /or 5 and their powers. So, here we need to use the fundamental theorem of arithmetic in expressing the underlying number (divisor) as a product of distinct primes. Lemma 3.2.1. Let 9 = ∑ 10 6 6