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Identical Digit Baffle (Posted on 2014-09-08) Difficulty: 2 of 5
Determine the largest five digit base-14 positive integer such that when multiplied by a single base-14 digit, the result is a six digit base-14 positive integer with all digits identical.

No Solution Yet Submitted by K Sengupta    
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Solution computer solution | Comment 1 of 2
DefDbl A-Z
Dim crlf$


Private Sub Form_Load()
 ChDir "C:\Program Files (x86)\DevStudio\VB\projects\flooble"
 Text1.Text = ""
 crlf$ = Chr(13) + Chr(10)
 Form1.Visible = True
 DoEvents
 
 For repdigit = 1 To 13
   repdnum = Int(repdigit * (14 ^ 6 - 1) / 13 + 0.5)
   For divsr = 1 To 13
     q = repdnum / divsr
     If Int(q) = q Then
       b14$ = base$(q, 14)
       If Len(b14$) = 5 Then
         Text1.Text = Text1.Text & b14$ & "(" & (q) & ")" & "   " & base$(repdnum, 14)
         Text1.Text = Text1.Text & "(" & (repdnum) & ")"
         Text1.Text = Text1.Text & "    " & base$(divsr, 14) & crlf
       End If
     End If
   Next
 Next
 
 Text1.Text = Text1.Text & crlf & " done"
End Sub

Function base$(n, b)
  v$ = ""
  n2 = n
  Do
    d = n2 Mod b
    n2 = n2 \ b
    v$ = Mid("0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ", d + 1, 1) + v$
  Loop Until n2 = 0
  base$ = v$
End Function

finds, (lines sorted by a separate step), this list of all such 5-digit base-14 integers:

base-14  decimal	 product
integer equivalent   base-14 decimal  multiplier
1964B    (64355)      111111(579195)     9
30303 (115839)     111111(579195)     5
30303 (115839)     222222(1158390)    A
34C98 (128710)     222222(1158390)    9
50505 (193065)     111111(579195)     3
50505 (193065)     222222(1158390)    6
50505 (193065)     333333(1737585)    9
50505 (193065)     444444(2316780)    C
60606 (231678)     222222(1158390)    5
60606 (231678)     444444(2316780)    A
69B52 (257420)     444444(2316780)    9
8539D (321775)     555555(2895975)    9
90909 (347517)     333333(1737585)    5
90909 (347517)     666666(3475170)    A
A0A0A (386130)     222222(1158390)    3
A0A0A (386130)     444444(2316780)    6
A0A0A (386130)     666666(3475170)    9
A0A0A (386130)     888888(4633560)    C
BA257 (450485)     777777(4054365)    9
C0C0C (463356)     444444(2316780)    5
C0C0C (463356)     888888(4633560)    A
D58A4 (514840)     888888(4633560)    9

So the largest is therefore D58A4, which when multiplied by 9 produces 888888. 


  Posted by Charlie on 2014-09-08 11:07:11
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