*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 | # | 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.