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 | # | min | max |
| 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.