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A pandigital number is a number which contains all ten digits at least once. We can call strictly pandigital those that contains exactly 10 digits

Pandigital multiples

There are 10! = 3,628,800 permutations of the 10 digits, and, disregarding those with a leadig 0, we obtain a total of 3,265,920 strictly pandigital numbers, from 1023456789 to 9876543210. (Let me drop the strict in the following...)

Among these numbers several have pandigital multiples too, like 1,287,609,354×7 = 9,013,265,478.

In the following table I summarized, for each k from 2 to 9, the number of pandigital numbers p such that the product p×k is pandigital as well. For each k, I report two examples of p, namely the smallest and the largest.

k#minmax
2 184,320 1,023,456,789 4,938,271,605
3 5,820 1,023,748,965 3,291,768,054
4 6,480 1,023,456,789 2,469,135,780
5 46,080 1,024,693,578 1,975,308,642
6 998 1,023,465,897 1,645,839,027
7 387 1,023,547,986 1,409,632,875
8 171 1,023,794,658 1,234,567,890
9 167 1,023,674,985 1,097,368,245

It is worth noting that there are several (12,289) of these pandigital numbers which have more than one pandigital multiple. In particular there are 8 numbers with 4 pandigital multiples and the 2 numbers (1,098,765,432 and 1,234,567,890) which have 5 pandigital multiples (both for k=2, 4, 5, 7, 8). Indeed, for p=1,098,765,432, we have

2 × p = 2,197,530,864
4 × p = 4,395,061,728
5 × p = 5,493,827,160
7 × p = 7,691,358,024
8 × p = 8,790,123,456.