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 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 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) :
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%.
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