- Polydivisible number
In

mathematics a**polydivisible number**is a number with digits*abcde...*that has the following properties :-# Its first digit

*a*is not 0.

# The number formed by its first two digits*ab*is a multiple of 2.

# The number formed by its first three digits*abc*is a multiple of 3.

# The number formed by its first four digits*abcd*is a multiple of 4.

# etc.For example, 345654 is a six-digit polydivisible 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.

The smallest base 10 polydivisible numbers with 1,2,3,4... etc. digits are

1, 10, 102, 1020, 10200, 102000, 1020005, 10200056, 102000564, 1020005640 OEIS|id=A078282

**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 are2,492 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 polydivisible number with "n" digits if there is a number between 10"k" and 10"k"+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 polydivisible number to an "n"-digit polydivisible 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 polydivisible number in this way, and as "n" becomes larger, the chances of being able to extend a given polydivisible number become smaller.

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

:$F(n)\; approx\; frac\{9\; imes\; 10^\{n-1\{n!\}$

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 polydivisible 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*** [

*http://jwilson.coe.uga.edu/emt725/Class/Lanier/Nine.Digit/nine.html The nine-digit problem and its solution*]

* [*http://www.filmshuren.nl/randomstuff/polydivisiblenumbers.txt A list of all 20,456 polydivisible numbers*]

*Wikimedia Foundation.
2010.*

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