At first I tried for 10-digit pandigital base values. The first time through, to save time, I went for even pandigitals to serve as 2k, as evidenced by the commented out code. When that didn't work, I left out the requirement for even number, and tried all 10-digit values for k. That didn't work.  Then I added an outer loop to try one duplicated digit, zero through 9, and achieved success. 

The multiples for the smallest successful k are:

  1     36492195078                                            

  2     72984390156                                            

  3    109476585234                                           

  4    145968780312                                           

  5    182460975390                                           

  6    218953170468                                           

As the numbers came out in order of the extra digit, I sorted the output (60 11-digit values of k work), bringing the smallest, with a duplicated 9, to the top. Each of these could serve as k:

36492195078

48602175913

48613021759

49021758613

49130217586

49219635078

53829197460

53829301746

53928301746

54601738293

54601739283

58829301746

59288301746

60174538293

60174539283

60174588293

60174592883

64820935179

64917935082

78036492195

78049219635

80253931746

80253994617

80317253946

80317392546

80317453926

80317539246

81725394603

81730253946

81739254603

81739302546

81745392603

81745393026

81746025393

81753924603

82064917935

82539301746

82539460173

82546017393

82601745393

83017253946

83017392546

83017453926

83017539246

85463728190

89465237801

89465237810

90175398246

93517906482

93780246551

93781024655

94652237801

94652237810

94655237801

94655237810

94785720163

94785721630

97539018246

98175390246

98246017539

DefDbl A-Z

Dim crlf$

Private Sub Form_Load()

 Form1.Visible = True

 

 Text1.Text = ""

 crlf = Chr$(13) + Chr$(10)

 digs$ = "1023456789"

 

 

 For xtra = 0 To 9

 s$ = digs + LTrim(Str(xtra)): h$ = s

 Do

   If Left(s, 1) > "0" Then

    ' If InStr("02468", Right(s, 1)) > 0 Then

       k = Val(s)

       good = 1

       For mult = 2 To 6

         ns$ = LTrim(Str(mult * k))

         For i = 1 To 10

           If InStr(ns, Mid(digs, i, 1)) = 0 Then good = 0: Exit For

         Next

         DoEvents

       Next

       If good Then Text1.Text = Text1.Text & k & crlf

   '  End If

   End If

   permute s

 Loop Until s = h

 Next xtra

 

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

  

End Sub