The program at the bottom of the post verifies there are a total of 2492 10-digit numbers meeting the divisibility criterion. It also lists the 19 cases where only 3 different digits are used, and that none require any fewer than 3 different digits.

     n                 diff digits

1800008010      3

2404024020      3

3000060000      3

4084080840      3

4440002400      3

4444020000      3

4808048040      3

5040000000      3

5404504050      3

6636000060      3

8048040840      3

8404088040      3

8820000000      3

8880004080      3

8880004800      3

8880060060      3

8888040000      3

9060009660      3

9660000060      3

2492 done

I notice that only one has a 5 in the fifth position. I imagine one reason for that is that one of the three used digits must be a zero in the last position so it might as well be used at position 5.

A modification of the program (If uCt < 4 Or uCt > 9) finds that there's also exactly one pandigital number also meeting the divisibility criterion: 3816547290. Of course that one has to have a 5 in the fifth position so as not to duplicate the terminal zero.

DefDbl A-Z

Dim crlf$, n, ct

Private Sub Form_Load()

 Form1.Visible = True

 Text1.Text = ""

 crlf = Chr(13) + Chr(10)

 

 n = 0

 

 addOn 1

 

 

 Text1.Text = Text1.Text & crlf & ct & " done"

  

End Sub

Sub addOn(wh)

 DoEvents

  saveN = n

  If wh = 1 Then st = 1 Else st = 0

  For newd = st To 9

    n = 10 * saveN + newd

    q = Int(n / wh): r = n - q * wh

    If r = 0 Then

      If wh = 10 Then

        uCt = 0: ReDim used(9)

        s$ = LTrim(Str(n))

        For i = 1 To Len(s)

          If used(Val(Mid(s, i, 1))) = 0 Then uCt = uCt + 1

          used(Val(Mid(s, i, 1))) = 1

        Next

        If uCt < 4 Then Text1.Text = Text1.Text & n & Str(uCt) & crlf

        ct = ct + 1

      Else

        addOn wh + 1

      End If

    End If

  Next

  n = saveN

End Sub