Polydivisible number

In mathematics a polydivisible number is a number with digits abcde... that has the following properties :-

1. Its first digit a is not 0.
2. The number formed by its first two digits ab is a multiple of 2.
3. The number formed by its first three digits abc is a multiple of 3.
4. The number formed by its first four digits abcd is a multiple of 4.
5. etc.

For example, 345654 is a six-digit polydivisble number, but 123456 is not, because 1234 is not a multiple of 4. Polydivisible numbers can be defined in any base- however, the numbers in this article are all in base 10, so permitted digits are 0 to 9.

Background

Polydivisible numbers are a generalisation of the following well-known problem in recreational mathematics :-

Arrange the digits 1 to 9 in order so that the first two digits form a multiple of 2, the first three digits form a multiple of 3, the first four digits form a multiple of 4 etc. and finally the entire number is a multiple of 9.

The solution to the problem is a nine-digit polydivisible number with the additional condition that it contains the digits 1 to 9 exactly once each. There are 2,492 nine-digit polydivisible numbers, but the only one that satisfies the additional condition is

381654729

How many polydivisible numbers are there?

If k is a polydivisible number with n-1 digits, then it can be extended to create a polydivisble number with n digits if there is a number between 10k and 10k+9 that is divisible by n. If n is less or equal to 10, then it is always possible to extend an n-1 digit polydivisble number to an n-digit polydivisble number in this way, and indeed there may be more than one possible extension. If n is greater than 10, it is not always possible to extend a polydivisble number in this way, and as n becomes larger, the chances of being able to extend a given polydivisble number become smaller.

On average, each polydivisble number with n-1 digits can be extended to a polydivisble number with n digits in 10/n different ways. This leads to the following estimate of the number of n-digit polydivisble numbers, which we will denote by F(n) :-

$F(n)\approx {\frac {9.10^{n-1}}{n!}}$

Summing over all values of n, this estimate suggests that the total number of polydivisible numbers will be approximately

${\frac {9(e^{10}-1)}{10}}\approx 19823$

In fact, this underestimates the actual number of polydivisible numbers by about 3%.

Counting polydivisible numbers

We can find the actual values of F(n) by counting the number of polydivisble numbers with a given length :-

Length n	F(n)	Estimate of F(n)		Length n	F(n)	Estimate of F(n)		Length n	F(n)	Estimate of F(n)
1	9	9		11	2225	2255		21	18	17
2	45	45		12	2041	1879		22	12	8
3	150	150		13	1575	1445		23	6	3
4	375	375		14	1132	1032		24	3	1
5	750	750		15	770	688		25	1	1
6	1200	1250		16	571	430
7	1713	1786		17	335	253
8	2227	2232		18	180	141
9	2492	2480		19	90	74
10	2492	2480		20	44	37

There are 20,456 polydivisible numbers altogether, and the longest polydivisible number, which has 25 digits, is :-

360 852 885 036 840 078 603 672 5

Related problems

Other problems involving polydivisible numbers include :-

• Finding polydivisible numbers with additional restrictions on the digits - for example, the longest polydivisible number that only uses even digits is

480 006 882 084 660 840 40

• Finding palindromic polydivisible numbers - for example, the longest palindromic polydivisible number is

300 006 000 03

• Enumerating polydivisible numbers in other bases.

External links

• The nine-digit problem and its solution