In mathematics a polydivisible number (or magic number) is a number with digits abcdef... 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.
For example, EX9876 is a six-digit polydivisible number, but 123456 is not, because 12345 is not a multiple of 5. 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 E.
There are 71823 polydivisible numbers, and the largest of them is 24-digit 6068,903468,50EX68,00E036,206464
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 10k and 10k+E 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 (e.g. for n=14, the chance is 90%, while for n=20, the chance is only 60%). 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 for F(n) :
F(n) ≈ (E*10^(n-1))/(n!)
Summing over all values of n, this estimate suggests that the total number of polydivisible numbers will be approximately
E*(e^10-1)/10 ≈ 72407
where e = 2.875236069822... is the base of natural logarithm.
There are about E*10^(n-1)/n! n-digit polydivisible numbers.
There are no E-digit polydivisible numbers using all the digits 1 to E exactly once. (hence there are also no 10-digit polydivisible numbers using all the digits 0 to E exactly once)