For every positive integer n, the set of all positive divisors of n becomes a partially ordered set if we use divisibility as the order relation.

The restriction of the divisibility relation to the set of all monic monomials ( in the given ring ) is a partial order, and thus makes this set to a poset.

The order ord < sub > n </ sub > a also divides λ ( n ), a value of the Carmichael function, which is an even stronger statement than the divisibility of φ ( n ).

*, the set of natural numbers ordered by divisibility, is not a well partial order: the prime numbers are an infinite antichain.

For example, if we order by divisibility, we end up with

The concept of infinite divisibility arises in different ways in philosophy, physics, economics, order theory ( a branch of mathematics ), and probability theory ( also a branch of mathematics ).

In another paper in 1897, Dedekind studied the lattice of divisors with gcd and lcm as operations, so that the lattice order is given by divisibility.

Other Möbius inversion formulas are obtained when different locally finite partially ordered sets replace the classic case of the natural numbers ordered by divisibility ; for an account of those, see incidence algebra.

Properties of the natural numbers related to divisibility, such as the distribution of prime numbers, are studied in number theory.

The divisibility relation among the P-smooth numbers, natural numbers whose prime factors all belong to the finite set P, gives these numbers the structure of a partially ordered set isomorphic to.

Digit sums and digital roots can be used for quick divisibility tests: a natural number is divisible by 3 or 9 if and only if its digit sum ( or digital root ) is divisible by 3 or 9, respectively.

This yields a formal definition of the field of Puiseux series: it is the direct limit of the direct system, indexed over the non-zero natural numbers n ordered by divisibility, whose objects are all ( the field of formal Laurent series, which we rewrite as

The base conversion divisibility test is a process that can be used to determine whether or not a certain ( positive ) natural number a can be divided evenly into a larger natural number b. It is the general case for the well-known test for divisibility by nine.

As a result of this increased factorability of the radix and its divisibility by a wide range of the most elemental numbers ( whereas ten has only two non-trivial factors: 2 and 5, with neither 3 nor 4 ), duodecimal representations fit more easily than decimal ones into many common patterns, as evidenced by the higher regularity observable in the duodecimal multiplication table.

Later ancient commentators such as Proclus ( 410 – 485 CE ) treated many questions about infinity as issues demanding proof and, e. g., Proclus claimed to prove the infinite divisibility of a line, based on a proof by contradiction in which he considered the cases of even and odd numbers of points constituting it.

Euclid devoted part of his Elements to prime numbers and divisibility, topics that belong unambiguously to number theory and are basic thereto ( Books VII to IX of Euclid's Elements ).

* Начало общего наибольшего делителя въ применении к теорiи делимости алгебраическихъ чиселъ ( Principle of greatest common divisor in the usage to the divisibility theory of algebraic numbers ) ( 1893 ),

Another test for divisibility is to separate a number into groups of two consecutive digits ( adding a leading zero if there is an odd number of digits ), and then add up the numbers so formed ; if the result is divisible by 11, the number is divisible by 11.

* Book 7 deals strictly with elementary number theory: divisibility, prime numbers, Euclid's algorithm for finding the greatest common divisor, least common multiple.

The fundamental problem of the calendar is the imperfect divisibility of whole numbers into an irrational number ( fitting whole days into a month ; fitting whole days or whole months into a year ).

Like Cullen numbers, Woodall numbers have many divisibility properties.

This might not be as simple as the divisibility tests for numbers like 3 or 5, and it might not be terribly practical, but it is simpler than the divisibility tests for other 3-digit numbers.

Given the divisibility test for 9, one might be tempted to generalize that all numbers divisible by 9 are also Harshad numbers.

On the divisibility of numbers

The somewhat confusing terminology and disagreement amongst geologists on where to draw what hierarchical boundaries, is due to the comparatively fine divisibility of time units as time approaches the present, and due to geological preservation that causes the youngest sedimentary geological record to be preserved over a much larger area and to reflect many more environments, than the older geological record.

In commutative algebra, one major focus of study is divisibility among polynomials.

Now, for distinct primes, these divisibility events are mutually independent because the candidate divisors are coprime ( a number is divisible by coprime divisors n and m if and only if it is divisible by nm, an event which occurs with probability 1 /( nm ).

Actual divisibility may be limited due to unavailability of cutting instruments, but its possibility of breaking into smaller pieces is infinite.

In arithmetic, for example, when multiplying by 9, using the divisibility rule for 9 to verify that the sum of digits of the result is divisible by 9 is a sanity test-it will not catch every multiplication error, however it's a quick and simple method to discover many possible errors.

In abstract algebra, two nonzero elements and of a ring are respectively called a left zero divisor and a right zero divisor if ; this is a partial case of divisibility in rings.

For this method, it is also necessary to check for divisibility by all primes that are less than c. Observations analogous to the preceding can be applied recursively, giving the Sieve of Eratosthenes.

*" Any-old-how ", in which the divisibility of " anything " ( as in " any old thing ") is mimicked with the usually indivisible " anyhow ".

Except for 11, all palindromic primes have an odd number of digits, because the divisibility test for 11 tells us that every palindromic number with an even number of digits is a multiple of 11.

It is based on ideas such as divisibility and congruence.

With this ordering, there is no point in testing for divisibility by four if the number has already been determined not divisible by two, and so on for three and any multiple of three, etc.

The divisibility is attributable to the alternate representation

Integral domains, non-trivial commutative rings where no two non-zero elements multiply to give zero, generalize another property of the integers and serve as the proper realm to study divisibility.

Visual logic is another theme that is used in Gothic Cathedrals, including progressive divisibility.

However, it can formulate individual instances of divisibility ; for example, it proves " for all x, there exists y: ( y + y

For example, for the problem " find all integers between 1 and 1, 000, 000 that are evenly divisible by 417 " a naive brute-force solution would generate all integers in the range, testing each of them for divisibility.

A central concept in number theory is " divisibility " ( example: 42 is divisible by 14 but not by 15 ).