Polydivisible number

In Mathematical, a polydivisible number is a Entier naturalness being written with the Chiffre S $a\; B\; C\; D\; E\; \backslash \; cdots\; \backslash ,$ which has the following properties:

1. Its first figure $a\; \backslash ,$ is not 0.

2. the number formed by its first two figures $a\; B\; \backslash ,$ is a multiple of 2.
3. the number formed by its first three figures $a\; B\; C\; \backslash ,$ is a multiple of 3.
4. the number formed by its first four figures $a\; B\; C\; D\; \backslash ,$ is a multiple of 4.
5. etc

For example, 345654 is a polydivisble number with six digits, but 123456 is not it, because 1234 are not a multiple of 4. The polydivisibles numbers can be defined in any bases. Notwithstanding, the numbers in this article are all in bases 10, thus, the figures from 0 to 9 are allowed.

Background

The polydivisibles numbers are a generalization of the very known problem following of entertaining Mathématiques:

To arrange the figures from 1 to 9 in an order where the first two figures form a multiple of 2, the first three figures form a multiple of 3, the first four figures form a multiple of 4 etc and finally, the integer is a multiple of 9.

The solution of the problem is a polydivisible number with nine digits with the additional condition which it contains the figures from 1 to 9 exactly once each one. There exist 2.492 polydivisibles numbers with nine digits, but the only one which satisfies the additional condition is

$381\; 654.729\; \backslash ,$

How much polydivisibles numbers exist?

If K is a polydivisible number with N -1 figures, then it can be wide to create a polydivisble number with N figures if there exists a number ranging between 10 K and 10 K +9 which is divisible by N . If N is lower or equal to 10, then it is always possible to extend a polydivisible number to N -1 figures in a polydivisible number with N figures of this manner, and it can exist more than one possible extension. If N is larger than 10, it is not always possible to extend a polydivisble number of this manner, and when N becomes larger, the chances to be able to extend a polydivisble number given become smaller.

On average, each polydivisble number of N -1 figures can be wide with a polydivisble number of N figure of 10 N different manners. This led to the following estimate of the number of polydivisibles numbers of N figures, which we will note F (N) :

$F\; (N)\; \backslash \; apex\; \backslash \; frac\; \{9.10^\; \{n-1\}\}\; \{N!\}$

By summoning all the values of N, this estimate suggests that the full number of polydivisibles numbers will be roughly

$\backslash \; frac\; \{9\; (e^\; \{10\}\; -\; 1)\}\{10\}\; \backslash \; apex\; 19823$

In fact, this approximation approaches the current number of numbers polydivisible of 3%.

To count the polydivisibles numbers

We can find the current values of F (N) by counting the number of polydivisbles numbers a given length:

There exist 20.456 polydivisibles numbers all units, and the longest polydivisible number, which has 25 digits, is:

$3\; 608.528.850\; 368.400.786\; 036.725\; \backslash ,$

Related problems

Other problems imply the polydivisibles numbers:

• Trouver polydivisibles numbers with additional restrictions on the figures - for example, the polydivisible number longest which does not contain that even figures is:

$48\; 000.688.208\; 466.084.040\; \backslash ,$

• To find the polydivisibles numbers palindromes - for example, the polydivisible number palindrome longest is

$30\; 000.600.003\; \backslash ,$

• To enumerate the polydivisibles numbers in the other bases.

External bond

• the problem of the nine digits and its solution

Random links:

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