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While much is known about product bundles comprised of different items, much less is known about multipacks—a product set comprised of multiple identical items (e.g. a 4-pack body washes). Using the context of multipacks, the authors propose a novel price divisibility effect, which suggests that a multipack's price that is easily divisible (vs. non-divisible) by the number of component items in the multipack will increase its purchase likelihood. For example, purchase likelihoods of a four-pack body wash multipack will be higher when its price is $16 (easily divisible by 4) versus $15.30 (non-divisible by 4). This occurs because a divisible versus non-divisible price shifts consumers' attention to the unit and creates a belief that each unit item in the multipack will be consumed quickly, which, in turn, helps justify purchasing multiple units. The authors report findings from 15 studies (including a field experiment), where they demonstrate the effect and its underlying mechanism and delineate several moderators and boundary conditions. This research contributes to several literature streams, including those on product bundling, multiple-unit pricing, product consumption, and numerical cognition. [ABSTRACT FROM AUTHOR]

Let Ω be a class of C*-algebras with the m-comparison property (respectively, the n-almost divisibility property, the weakly (k, n)-divisibility property). We show that any infinite-dimensional simple unital C*-algebra in the class GTA Ω (the class of C*-algebras which can be generalized tracially approximated by the C*-algebras in Ω ) has m-comparison (respectively, is (2 n + 1) -almost divisible, is weakly (k, 2n)-divisible). [ABSTRACT FROM AUTHOR]

Let a, b and n be positive integers and let $ S=\{x_1,\ldots, x_n\} $ S = { x 1 , ... , x n } be a set of n distinct positive integers. For $ x\in S $ x ∈ S , one defines $ G_{S}(x)=\{y\in S: y \lt x, y|x \ {\rm and} \ (y|z|x, z\in S)\Rightarrow z\in \{y,x\}\} $ G S (x) = { y ∈ S : y < x , y | x and (y | z | x , z ∈ S) ⇒ z ∈ { y , x } }. For any arithmetic function f and any positive integer x, we define the power arithmetic function $ f^a $ f a by $ f^a(x)=(f(x))^a $ f a (x) = (f (x)) a . We denote by $ (f^a[S]) $ (f a [ S ]) the $ n\times n $ n × n power matrix having $ f^a $ f a evaluated at the least common multiple of $ x_i $ x i and $ x_j $ x j as its $ (i,j) $ (i , j) -entry. Denote by $ {\rm lcm}(S) $ lcm (S) the least common multiple of all the elements of S and by $ |T| $ | T | the number of elements of any finite set T. In this paper, we show that if $ a|b $ a | b , S is gcd closed (i.e. $ \gcd (x_i, x_j)\in S $ gcd (x i , x j) ∈ S for all integers i and j with $ 1\le i, j\le n $ 1 ≤ i , j ≤ n) with $ \max _{x\in S}\{|G_S (x)|\}=1 $ max x ∈ S { | G S (x) | } = 1 and f is a multiplicative function such that $ (f^a]S]) $ (f a ] S ]) is nonsingular and $ (f*\mu)(d)\in \mathbb {Z} $ (f ∗ μ) (d) ∈ Z whenever $ d|{\rm lcm}(S) $ d | lcm (S) and $ f(y)|f(x) $ f (y) | f (x) whenever $ y|x $ y | x and $ y, x\in S $ y , x ∈ S , where $ f*\mu $ f ∗ μ is the Dirichlet convolution of f and the Möbius function μ, then the ath power matrix $ (f^a[S]) $ (f a [ S ]) divides the bth power matrix $ (f^b[S]) $ (f b [ S ]) in the ring of $ n\times n $ n × n matrices over the integers. Our result extend a theorem of S.F. Hong [Divisibility properties of power GCD matrices and power LCM matrices. Linear Algebra Appl. 2008;428:1001–1008] and that of G.Y. Zhu and M. Li [On the divisibility among power LCM matrices on gcd-closed sets. Bull Aust Math Soc. 2023;107:31–39]. [ABSTRACT FROM AUTHOR]

The Dushnik–Miller dimension of a poset P is the least d for which P can be embedded into a product of d chains. Lewis and Souza isibility order on the interval of integers [ N / κ , N ] is bounded above by κ (log κ) 1 + o (1) and below by Ω ((log κ / log log κ) 2) . We improve the upper bound to O ((log κ) 3 / (log log κ) 2). We deduce this bound from a more general result on posets of multisets ordered by inclusion. We also consider other divisibility orders and give a bound for polynomials ordered by divisibility. [ABSTRACT FROM AUTHOR]

In this article, we deal with the algebraic structures of the partial orders induced by non-divisible t-norms. We first give a condition for the partial orders induced by non-divisible t-norms being a meet semi-lattice. Then, we show the condition for the partial orders induced by right-continuous t-norms being a lattice, which is very useful for constructing new lattice structures. Finally, an illustrative example is presented. Our results go a further step towards answering the first open problem raised by Karaçal and Kesicioğlu (Kybernetika 47:300-314, 2011). [ABSTRACT FROM AUTHOR]

This paper formalizes problems 41, 92, 121–123, 172, 182, 183, 191, 192 and 192a from "250 Problems in Elementary Number Theory" by Wacław Sierpiński [8]. [ABSTRACT FROM AUTHOR]

In, we started the investigation of pseudo MV-algebras with square roots. In the present paper, the main aim is to continue to study the structure of pseudo MV-algebras with square roots focusing on their new characterizations. The paper is divided into two parts. In the first part, we investigate the relationship between a pseudo MV-algebra with square root and its corresponding unital l-group in the scene of two-divisibility. We characterize strict and nonstrict square roots, and we describe square roots on strongly (H, 1)-perfect pseudo MV-algebras. In the present second part, we find some conditions under which a particular class of pseudo MV-algebras can be embedded into pseudo MV-algebras with square roots. We compare both the concepts of a strict square root closure and a square root closure of a pseudo MV-algebra. We show that each MV-algebra has a square root closure. Finally, using the square root of individual elements of a pseudo MV-algebra, we find the greatest subalgebra of a special pseudo MV-algebra with weak square root. [ABSTRACT FROM AUTHOR]

In, we started the investigation of pseudo MV-algebras with square roots. In the present paper, we continue to study the structure of pseudo MValgebras with square roots focusing on their new characterizations. The paper is divided into two parts. In the present first part, we investigate the relationship between a pseudo MV-algebra with square root and its corresponding unital ℓ-group in the scene of two-divisibility. We characterize strict and non-strict square roots and we describe square roots on strongly (H, 1)-perfect pseudo MV-algebras. In the second part, we find some conditions under which a particular class of pseudo MV-algebras can be embedded into pseudo MV-algebras with square roots. We investigate the concepts of a strict square root closure and a square root closure of a pseudo MV-algebra, and we compare both notions. We show that each MV-algebra has a square root closure. Finally, using the square root of individual elements of a pseudo MV-algebra, we find the greatest subalgebra of a special pseudo MV-algebra with weak square root. [ABSTRACT FROM AUTHOR]

We study divisibility properties of a set { f 1 (U n (s)) , ... , f m (U n (s)) } , where f 1 , ... , f m are polynomials in s variables over Z and U n (s) is a point picked uniformly at random from the set { 1 , ... , n } s. We show that, as n → ∞ , the GCD and the suitably normalized LCM of this set converge in distribution to a.s. finite random variables under mild assumptions on f 1 , ... , f m. Our approach is based on the known fact that the uniform distribution on { 1 , ... , n } converges to the Haar measure on the ring Z ˆ of profinite integers, combined with the Lang–Weil bounds and tools from probability theory. [ABSTRACT FROM AUTHOR]

Motion—and, in particular, local motion or change in location—plays a central role in Kenelm Digby’s natural philosophy and in his arguments for the immateriality of the soul. Despite this, Digby’s account of what motion consists in has yet to receive much scholarly attention. In this paper, I advance a novel interpretation of Digby on motion. According to it, Digby holds that for a body to move is for it to divide from and unify with other bodies. This is a view of motion—as change in relations of parthood—that Alison Peterman attributes to Digby’s contemporary and acquaintance, Margaret Cavendish. Having shown that Digby’s presentation of the view predates Cavendish’s by more than a decade, I make a case that Digby’s work influenced Cavendish’s on this topic. In developing and defending my reading, I consider to what extent the Digbean account of motion and the arguments for it accord with the ideals of the mechanical philosophy emerging in the early modern period.

We say that a C*-algebra is nowhere scattered if none of its quotients contains a minimal open projection. We characterize this property in various ways, by topological properties of the spectrum, by divisibility properties in the Cuntz semigroup, by the existence of Haar unitaries for states, and by the absence of nonzero ideal-quotients that are elementary, scattered or type I. Under the additional assumption of real rank zero or stable rank one, we show that nowhere scatteredness implies even stronger divisibility properties of the Cuntz semigroup. [ABSTRACT FROM AUTHOR]

This paper establishes certain formulas for p-adic valuation of Stirling numbers of the second kind S(pn, k) where p is a prime and some related classes. The parity of k also affects the p-adic valuation of S(n, k) if k is divisible by p. In fact, vp(S(pN, kp)) ≥ 5 if k is even. The congruence properties of S(pn, k) (mod pN) depend on the sum of the p-adic digits of k when k is not a multiple of p. [ABSTRACT FROM AUTHOR]

Understanding divisibility at the primary school level is a strong predictor of students’ mathematical achievements in secondary education. To correctly measure students’ understanding and achievements, a valid and reliable test is needed. This research focuses on the construction of valid and reliable test for the divisibility area studied at the primary school level. After constructing three pilot tests according to learning outcomes and standards for divisibility, and qualitative validation, tests were distributed in six primary schools, with 380 participating students (ages 12 – 13). The results were used for reliability and quantitative item analysis, and the final version of the test, which covered standards of students’ achievement and had all items of appropriate difficulty and discriminative validity, was created. This test can be used by mathematics teachers in classrooms but also in large scale testing, like state or international testing. [ABSTRACT FROM AUTHOR]

This research seeks to examine, first, whether and why consumers perceive divisible versus indivisible numbers differently and, second, how such divergent perceptions influence consumer preferences for marketer-created entities associated with divisible versus indivisible numbers. Integrating insights from two different literatures—numerical cognition and loneliness—we propose and find that numbers perceived to be divisible (vs. indivisible) are viewed as having more "connections" and are therefore deemed to be less lonely. Building on these findings and the literature on compensatory consumption, we then propose and demonstrate that a temporary feeling of loneliness increases participants' relative preference for various targets—products, attributes, and prices—associated with divisible (vs. indivisible) numbers, which are perceived to be relatively more connected and less lonely. It merits mention that our findings are triangulated across a wide variety of numbers, different product categories, and multiple operationalizations of loneliness. [ABSTRACT FROM AUTHOR]

This notes presents results related to divisibility or multiplicity between two numbers in the class of integers called smoothly undulating numbers of the type uz[n]. The main result is to characterize and display types of divisors of some types of numbers uz[n], and we show an algorithm to determine the greatest common divisor between two numbers uz[n].

A graph G is divisible by a graph H if the characteristic polynomial of G is divisible by that of H. In this paper, a necessary and sufficient condition for recursive graphs to be divisible by a path is used to show that the H-shape graph P 2 , 2 ; n − 4 2 , n − 7 , known to be (for n large enough) the minimizer of the spectral radius among the graphs of order n and diameter n − 5 , is determined by its adjacency spectrum if and only if n ≠ 10 , 13 , 15. [ABSTRACT FROM AUTHOR]

There exist several versions of discrete counterpart of continuity in the framework of finite chains, e.g., the smoothness, the divisibility, intermediate-value property and the 1-Lipschitz property. In this paper, we first discuss the relationships among the smoothness, divisibility, intermediate-value property and 1-Lipschitz property. Second, we present complete characterizations of divisible associative aggregation operations on finite chains. [ABSTRACT FROM AUTHOR]

In this paper, we first prove two new identities concerning the q-integers and then for these integers we derive several divisibility properties based on these identities and on properties of the q-integers. [ABSTRACT FROM AUTHOR]

In this paper problems 48, 80, 87, 89, and 124 from [7] are formalized, using the Mizar formalism [1], [2], [4]. The work is natural continuation of [5] and [3] as suggested in [6]. [ABSTRACT FROM AUTHOR]

We give necessary and sufficient conditions for the divisibility of two finite geometric series G n (x) = 1 + x + x 2 + ⋯ + x n - 1 over a field of characteristic zero. [ABSTRACT FROM AUTHOR]

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We prove that the modal logic of abelian groups with the accessibility relation of being isomorphic to a subgroup is S 4.2 . [ABSTRACT FROM AUTHOR]

Divisibility or partial rescission of contract is one of the guarantees for execution of contracts widely accepted by majority of legal institutions and international systems in developed countries. This option helps partial execution of contract instead of total revocation or complete enforcement of the contract. Thanks to its special qualities, installment contract has been paid due attention in terms of divisibility to the extent that the United Nations Convention on Contracts for the International Sale of Goods has enacted certain articles about it, emphasizing divisibility of installment contract with approach highlighting its efficiency. Since there is no such solution in Iranian law, particularly in terms of installment contract, in this paper we have tried to first present a precise definition of installment contract on one hand, and present solutions from Imamiyah jurisprudence, with Imam Khomeini’s opinion included, for divisibility of installment contract. On the other hand, we have tried to investigate some criteria of economic analysis approach to prove that accepting divisibility will boost the efficiency of installment contract.

In this paper we present an interesting result in elementary number theory, known as Zsigmondy’s theorem. We discuss some of its consequences and show its applications on the series of examples. [ABSTRACT FROM AUTHOR]

It is known that a C^*-algebra with the Global Glimm Property is nowhere scattered (it has no elementary ideal-quotients), and the Global Glimm Problem asks if the converse holds. We provide a new approach to this long-standing problem by showing that a C^*-algebra has the Global Glimm Property if and only if it is nowhere scattered and its Cuntz semigroup is ideal-filtered (the Cuntz classes generating a given ideal are downward directed) and has property (V) (a weak form of being sup-semilattice ordered). We show that ideal-filteredness and property (V) are automatic for C^*-algebras that have stable rank one or real rank zero, thereby recovering the solutions to the Global Glimm Problem in these cases. We also use our approach to solve the Global Glimm Problem for new classes of C^*-algebras. [ABSTRACT FROM AUTHOR]

Several non-standard problems are regarded and the possibilities of simple approaching to their solution are regarded and discussed.

Let $ a, b $ and $ n $ be positive integers and let $ S $ be a set consisting of $ n $ distinct positive integers $ x_1, ..., x_{n-1} $ and $ x_n $. Let $ (S^a) $ (resp. $ [S^a] $) denote the $ n\times n $ matrix having $ \gcd(x_i, x_j)^a $ (resp. $ {\rm lcm}(x_i, x_j)^a $) as its $ (i, j) $-entry. For any integer $ x\in S $, if $ (y < x, y|z|x \ {\rm and} \ y, z\in S)\Rightarrow z\in\{y, x\} $, then $ y $ is called a greatest-type divisor of $ x $ in $ S $. In this paper, we establish some results about the divisibility between $ (S^a) $ and $ (S^b) $, between $ (S^a) $ and $ [S^b] $ and between $ [S^a] $ and $ [S^b] $ when $ a|b $, $ S $ is gcd closed (i.e., $ \gcd(x_i, x_j)\in S $ for all $ 1\le i, j\le n $), and $ \max_{x\in S}\{|\{y\in S: y \ \text{is a greatest-type divisor of} \ x \ {\rm in} \ S\}|\} = 2 $.

In this work, we present a new general recursive divisibility test by prime numbers, involving a sequence based on the digits of the decimal base, we generalize the famous criterion for divisibility by 3. Finally, as an example of applications, we note that the some specific known cases remain valid. [ABSTRACT FROM AUTHOR]

Let $a,b$ and n be positive integers and let $S=\{x_1, \ldots , x_n\}$ be a set of n distinct positive integers. For ${x\in S}$ , define $G_{S}(x)=\{d\in S: d. Denote by $[S^a]$ the $n\times n$ matrix having the a th power of the least common multiple of $x_i$ and $x_j$ as its $(i,j)$ -entry. We show that the b th power matrix $[S^b]$ is divisible by the a th power matrix $[S^a]$ if $a\mid b$ and S is gcd closed (that is, $\gcd (x_i, x_j)\in S$ for all integers i and j with $1\le i, j\le n$) and $\max _{x\in S} \{|G_S (x)|\}=1$. This confirms a conjecture of Shaofang Hong ['Divisibility properties of power GCD matrices and power LCM matrices', Linear Algebra Appl. 428 (2008), 1001–1008]. [ABSTRACT FROM AUTHOR]

Let X be a projective variety and D be an effective Cartier divisor on X. In Int J Math 32(12), Paper No. 2140015, 2021, He and Ru author introduced the notion of the Nevanlinna hyperbolicity for the pair (X, D). In this paper, we derive some further properties for the Nevanlinna hyperbolic pairs, as well as provide multiple examples of the Nevanlinna hyperbolic pairs (X, D), mainly for certain blowup varieties. [ABSTRACT FROM AUTHOR]

An integral domain D is called an irreducible-divisor-finite domain (IDF-domain) if every nonzero element of D has finitely many irreducible divisors up to associates. The study of IDF-domains dates back to the seventies. In this paper, we investigate various aspects of the IDF property. In 2009, P. Malcolmson and F. Okoh proved that the IDF property does not ascend from integral domains to their corresponding polynomial rings, answering a question posed by D. D. Anderson, D. F. Anderson, and the second author two decades before. Here we prove that the IDF property ascends in the class of PSP-domains, generalizing the known result (also by Malcolmson and Okoh) that the IDF property ascends in the class of GCD-domains. We put special emphasis on IDF-domains where every nonunit is divisible by an irreducible, which we call TIDF-domains, and we also consider PIDF-domains, which form a special class of IDF-domains introduced by Malcolmson and Okoh in 2006. We investigate both the TIDF and the PIDF properties under taking polynomial rings and localizations. We also delve into their behavior under monoid domain and D + M constructions. [ABSTRACT FROM AUTHOR]

In this paper we confirm three conjectures of Z.-W. Sun on determinants. We first show that any odd integer n>3 divides the determinant (i 2 + d j 2) i 2 + d j 2 n 0 ≤ i , j ≤ (n − 1) / 2 , where d is any integer and (⋅ n) is the Jacobi symbol. Then we prove some divisibility results concerning |(i + dj)n|0≤i,j≤n−1 and |(i2 + dj2)n|0≤i,j≤n−1, where d ≠ 0 and n>2 are integers. Finally, for any odd prime p and integers c and d with p ∤ c d , we determine completely the Legendre symbol S c (d , p) p , where S c (d , p) := i 2 + d j 2 + c p 1 ≤ i , j ≤ (p − 1) / 2 . [ABSTRACT FROM AUTHOR]

In this paper, we first give the Pascal’s identity for Pellnomial coefficients and then we show that the Pellnomial coefficients are integers. We obtain that the product of r consecutive Pell numbers is divisible by the Pell analog of r!. Also, we introduce the divisibility theorems between Pell numbers and Pellnomial coefficients. Furthermore, we first define Pell–Catalan numbers and then we derive two formulas for presenting Pell–Catalan numbers. [ABSTRACT FROM AUTHOR]

Let U = (U n) n ≥ 0 be a Lucas sequence and, for every prime number p , let ρ U (p) be the rank of appearance of p in U , that is, the smallest positive integer k such that p divides U k , whenever it exists. Furthermore, let d be an odd positive integer. Under some mild hypotheses, we prove an asymptotic formula for the number of primes p ≤ x such that d divides ρ U (p) , as x → + ∞. [ABSTRACT FROM AUTHOR]

This paper revisits one of the least understood elements of Derrida's corpus: his sustained critique of Lacan's conception of the letter operative in the unconscious. Showing where and how this critique has been misconstrued, the paper demonstrates that the ultimate significance of Derrida's intervention lies in how it brings forward the uncritical conception of heterogeneity found in Lacan. In this way, Derrida's engagement with Lacan, from 'Positions' all the way up to the late seminars on The Beast and the Sovereign, sheds crucial light on how the core 'ultra-transcendental' structures of deconstruction are to be understood. Above all, it allows us to see how the ultra-transcendental logic of différance entails the thinking of finitude Derrida came to call la vie la mort, life death. [ABSTRACT FROM AUTHOR]

We give a characterization for the integers $ n \geq 1 $ such that the Fibonomial coefficient $ {pn \choose n}_F $ is divisible by $ p $ for any prime $ p \neq 2, 5 $. Then we use it to calculate asymptotic formulas for the number of positive integers $ n \leq x $ such that $ p \mid {pn \choose n}_F $. This completes the study on this problem for all primes $ p $.

The divisibility of the second-order minors of the numerators of transfer matrices by their minimal denominators for cyclic fractional linear systems is analyzed. It is shown that all nonzero second-order minors of the numerators of the transfer matrices are divisible by their minimal denominators if and only if the system matrices of fractional standard and descriptor linear systems are cyclic. The theorems are illustrated by examples of fractional standard and descriptor linear systems.

Divisibility is an old topic that to this day intrigues and fascinates researchers and scholars. Several rules are well-known in particular the divisibility by eleven, since, for example, a palindrome with an even number of digits is divisible by eleven. In current times, divisibility has its applications, for example, in cryptography. Here, in this paper, we will show that applying two somewhat intuitive procedures to the lines of Pascal's triangle shall always yield numbers divisible by eleven. Illustrative examples are presented.

This paper reports on the formalization of ten selected problems from W. Sierpinski's book "250 Problems in Elementary Number Theory" [5] using the Mizar system [4], [1], [2]. Problems 12, 13, 31, 32, 33, 35 and 40 belong to the chapter devoted to the divisibility of numbers, problem 47 concerns relatively prime numbers, whereas problems 76 and 79 are taken from the chapter on prime and composite numbers. [ABSTRACT FROM AUTHOR]

In this paper problems 17, 18, 26, 27, 28, and 98 from [9] are formalized, using the Mizar formalism [8], [2], [3], [6]. [ABSTRACT FROM AUTHOR]

Let E k be the normalized Eisenstein series of weight k on SL 2 (Z) . Let φ k be the polynomial that encodes the j-invariants of non-elliptic zeros of E k . In 2001, Gekeler observed that the polynomials φ k seem to be irreducible (and verified this claim for k ≤ 446 ). We show that φ k is irreducible for infinitely many k. [ABSTRACT FROM AUTHOR]

Let a , b and n be positive integers with n ≥ 2 , f be an integer-valued arithmetic function, and the set S = { x 1 , ... , x n } of n distinct positive integers be a divisor chain such that x 1 | x 2 | ⋯ | x n. We first show that the matrix (f a (S)) having f evaluated at the a th power (x i , x j) a of the greatest common divisor of x i and x j as its i , j -entry divides the GCD matrix (f b (S)) in the ring M n (Z) of n × n matrices over integers if and only if f b − a (x 1) ∈ Z and (f a (x i) − f a (x i − 1)) divides (f b (x i) − f b (x i − 1)) for any integer i with 2 ≤ i ≤ n. Consequently, we show that the matrix (f a [ S ]) having f evaluated at the a th power [ x i , x j ] a of the least common multiple of x i and x j as its i , j -entry divides the matrix (f b [ S ]) in the ring M n (Z) if and only if f b − a (x n) ∈ Z and (f a (x i) − f a (x i − 1)) divides (f b (x i) − f b (x i − 1)) for any integer i with 2 ≤ i ≤ n. Finally, we prove that the matrix (f a (S)) divides the matrix (f b [ S ]) in the ring M n (Z) if and only if f a (x 1) | f b (x i) and (f a (x i) − f a (x i − 1)) | (f b (x i) − f b (x i − 1)) for any integer i with 2 ≤ i ≤ n. Our results extend and strengthen the theorems of Hong obtained in 2008. [ABSTRACT FROM AUTHOR]

Let a and n be positive integers and let S = { x 1 , ... , x n } be a set of n distinct positive integers. For x ∈ S , one defines G S (x) = { d ∈ S : d < x , d | x   and   (d | y | x , y ∈ S) ⇒ y ∈ { d , x } }. We denote by (S a) (respectively, [ S a ]) the n × n matrix having the a th power of the greatest common divisor (respectively, the least common multiple) of x i and x j as its (i , j) -entry. In this paper, we show that for arbitrary positive integers a and b with a | b , the b th power matrices (S b) and [ S b ] are both divisible by the a th power matrix (S a) if S is a gcd-closed set (i.e. gcd (x i , x j) ∈ S for all integers i and j with 1 ≤ i , j ≤ n) such that max x ∈ S { | G S (x) | } = 1. This confirms two conjectures of Shaofang Hong proposed in 2008. [ABSTRACT FROM AUTHOR]

Divisibility, which can mean dividing an integer by another integer without a remainder, also includes finding the remainder in division. In teaching mathematics; odd, even, prime, etc. states of numbers are important for integers and related subjects, it plays a role in determining many of a number's properties. The divisibility rules of certain numbers are known to students. However, the rules of some numbers are determined by the factors of that number. For example, in the divisibility rule by 12 states, "If the number is divisible by both 3 and 4, it is also divisible by 12." Our aim in this paper is to express the divisibility rule by 12 differently and independently from its multipliers (3 and 4) and to create a completely new algorithm suitable in order to achieve our goal. For our purpose, firstly, it was desired to create a systematic order with numbers that are a whole multiple of 12. Then, a new divisibility algorithm was developed by working with division including remainders. It has been proven that the algorithm works flawlessly in studies using multiple different sized numbers and their resulting mathematical explanation and proofs. [ABSTRACT FROM AUTHOR]