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.
Table of contents |
2 How many polydivisible numbers are there ? 3 Counting polydivisible numbers 4 Related problems 5 External links |
Polydivisible numbers are a generalisation of the following well-known problem in recreational mathematics :-
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) :-
Background
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 isHow many polydivisible numbers are there ?
Summing over all values of n, this estimate suggests that the total number of polydivisible numbers will be approximately
In fact, this underestimates the actual number of polydivisible numbers by about 3%.
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
- Finding palindromic polydivisible numbers - for example, the longest palindromic polydivisible number is
- Enumerating polydivisible numbers in other bases.
External links