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 Multiples with 3 (Posted on 2017-01-31)
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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 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)

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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