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Multiples with 3 (Posted on 2017-01-31) Difficulty: 3 of 5
The first three multiples of 1153 all contain the digit 3: 1153, 2306, 3459. But its next multiple 4612 does not.

What is the smallest integer whose first four multiples all contain a 3?

What about 5 or 6 multiples?

No Solution Yet Submitted by Brian Smith    
Rating: 5.0000 (1 votes)

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Solution computer solution, taken beyond | Comment 1 of 3
The smallest integer whose first four multiples all contain a 3 is 1183, where the multiples are 1183, 2366, 3549, 4732.

For 5 and 6 multiples, etc. refer to the following table (3465 and 7673).

The program considered only numbers of two or more digits, so the reporting of 13 as the first with a single multiple was removed manually, as that of course would be 3 itself, a single digit number.

The reporting is for the lowest number that has exactly that many initial multiples as listed--no more, no less. So some lower number may appear later, representing a larger number of multiples, and therefore being a better answer for the lower number of multiples. As a result:

Some numbers of multiples are left out, such as 9, as 10 has a smaller number than the first number that has exactly 9 multiples and fail on the tenth. The same is true for example with 21, which serves also for 19 and 20. Despite that, for example, there shows up 53465 as the first number whose first 7 multiples, but not 8, have a 3; however 7673 has its first 8 multiples contain a 3, and therefore its first 7 also as well as for 6, and therefore fits as one of the asked-for answers, and 65913, shown for 11 multiples, could also serve for 7 or 10.

2 153
     153
     306
3 1153
     1153
     2306
     3459
4 1183
     1183
     2366
     3549
     4732
5 3465
     3465
     6930
     10395
     13860
     17325
6 34566
     34566
     69132
     103698
     138264
     172830
     207396
7 53465
     53465
     106930
     160395
     213860
     267325
     320790
     374255
8 7673
     7673
     15346
     23019
     30692
     38365
     46038
     53711
     61384
10 173277
     173277
     346554
     519831
     693108
     866385
     1039662
     1212939
     1386216
     1559493
     1732770
11 65913
     65913
     131826
     197739
     263652
     329565
     395478
     461391
     527304
     593217
     659130
     725043
12 76923
     76923
     153846
     230769
     307692
     384615
     461538
     538461
     615384
     692307
     769230
     846153
     923076
13 394659
     394659
     789318
     1183977
     1578636
     1973295
     2367954
     2762613
     3157272
     3551931
     3946590
     4341249
     4735908
     5130567
14 344717
     344717
     689434
     1034151
     1378868
     1723585
     2068302
     2413019
     2757736
     3102453
     3447170
     3791887
     4136604
     4481321
     4826038
15 257673
     257673
     515346
     773019
     1030692
     1288365
     1546038
     1803711
     2061384
     2319057
     2576730
     2834403
     3092076
     3349749
     3607422
     3865095
16 2769233
     2769233
     5538466
     8307699
     11076932
     13846165
     16615398
     19384631
     22153864
     24923097
     27692330
     30461563
     33230796
     36000029
     38769262
     41538495
     44307728
17 232767
     232767
     465534
     698301
     931068
     1163835
     1396602
     1629369
     1862136
     2094903
     2327670
     2560437
     2793204
     3025971
     3258738
     3491505
     3724272
     3957039
18 7692391
     7692391
     15384782
     23077173
     30769564
     38461955
     46154346
     53846737
     61539128
     69231519
     76923910
     84616301
     92308692
     100001083
     107693474
     115385865
     123078256
     130770647
     138463038
21 2307767
     2307767
     4615534
     6923301
     9231068
     11538835
     13846602
     16154369
     18462136
     20769903
     23077670
     25385437
     27693204
     30000971
     32308738
     34616505
     36924272
     39232039
     41539806
     43847573
     46155340
     48463107
22 3076923
     3076923
     6153846
     9230769
     12307692
     15384615
     18461538
     21538461
     24615384
     27692307
     30769230
     33846153
     36923076
     39999999
     43076922
     46153845
     49230768
     52307691
     55384614
     58461537
     61538460
     64615383
     67692306
23 6923313
     6923313
     13846626
     20769939
     27693252
     34616565
     41539878
     48463191
     55386504
     62309817
     69233130
     76156443
     83079756
     90003069
     96926382
     103849695
     110773008
     117696321
     124619634
     131542947
     138466260
     145389573
     152312886
     159236199
     
DefDbl A-Z
Dim crlf$, smallest(50)


Private Sub Form_Load()
 Form1.Visible = True
 
 Text1.Text = ""
 crlf = Chr$(13) + Chr$(10)
 
 For i = 1 To 50
   smallest(i) = 99999999999#
 Next

 For st = 1 To 1000000
   sts$ = LTrim(Str(st))
   For i = 0 To Len(sts)
    DoEvents
     ns$ = Left(sts, i) + "3" + Mid(sts, i + 1)
     n = Val(ns): nm = n
     For m = 2 To 50
       nm = nm + n
       nms$ = LTrim(Str(nm))
       If InStr(nms, "3") = 0 Then Exit For
     Next
     m = m - 1
     If n < smallest(m) Then
         smallest(m) = n
     End If
   Next
 Next
 
 For m = 1 To 50
    If smallest(m) < 99999999999# Then
     DoEvents
       Text1.Text = Text1.Text & m & Str(smallest(m)) & crlf
       For j = 1 To m
         Text1.Text = Text1.Text & "     " & j * smallest(m) & crlf
       Next j
     End If
 Next m
 

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

Edited on January 31, 2017, 9:43 pm
  Posted by Charlie on 2017-01-31 12:59:29

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