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A brute force approach to find multipliers for the first part of Peter Winkler's math puzzle 'Zeros, ones or twos', in his book _Mathematical Puzzles_
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# A horrible awful brute-force approach to Peter Winkler's 'Zeroes, ones and twos' problem, part 1 | |
(1..100).each do |p| | |
found = false | |
q = 0 | |
until found | |
q += 1 | |
product = p * q | |
if !(product.to_s =~ /[^01]/) || q > 1000000000 | |
found = true | |
p "#{p} * #{q} = #{product}" | |
end | |
end | |
end |
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# A slightly less brute-force approach to Peter Winkler's 'Zeroes, ones and twos' problem, part 1 | |
# this time it only checks each multiplier for the remaining undiscovered numbers | |
require 'pp' | |
pairs = {} | |
my_range = (1..1000) | |
partners = my_range.to_a | |
q = 0 | |
until partners.empty? || q > 100000000000 | |
q += 1 | |
hits , misses = partners.partition {|p| !((p * q).to_s =~ /[^01]/)} | |
hits.each do |p| | |
pairs[p]=[q,p*q] | |
p "#{pairs.length}: #{p} * #{q} = #{p*q}" | |
end | |
partners = misses | |
end | |
pp pairs.sort |
And while we're at it, here are the numbers such that the product has no digits larger than 4
:
[[1, [1, 1]],
[2, [1, 2]],
[3, [1, 3]],
[4, [1, 4]],
[5, [2, 10]],
[6, [2, 12]],
[7, [2, 14]],
[8, [3, 24]],
[9, [16, 144]],
[10, [1, 10]],
[11, [1, 11]],
[12, [1, 12]],
[13, [1, 13]],
[14, [1, 14]],
[15, [2, 30]],
[16, [2, 32]],
[17, [2, 34]],
[18, [8, 144]],
[19, [6, 114]],
[20, [1, 20]],
[21, [1, 21]],
[22, [1, 22]],
[23, [1, 23]],
[24, [1, 24]],
[25, [4, 100]],
[26, [4, 104]],
[27, [9, 243]],
[28, [4, 112]],
[29, [7, 203]],
[30, [1, 30]],
[31, [1, 31]],
[32, [1, 32]],
[33, [1, 33]],
[34, [1, 34]],
[35, [4, 140]],
[36, [4, 144]],
[37, [3, 111]],
[38, [3, 114]],
[39, [6, 234]],
[40, [1, 40]],
[41, [1, 41]],
[42, [1, 42]],
[43, [1, 43]],
[44, [1, 44]],
[45, [32, 1440]],
[46, [5, 230]],
[47, [3, 141]],
[48, [3, 144]],
[49, [7, 343]],
[50, [2, 100]],
[51, [2, 102]],
[52, [2, 104]],
[53, [4, 212]],
[54, [6, 324]],
[55, [2, 110]],
[56, [2, 112]],
[57, [2, 114]],
[58, [4, 232]],
[59, [7, 413]],
[60, [2, 120]],
[61, [2, 122]],
[62, [2, 124]],
[63, [7, 441]],
[64, [5, 320]],
[65, [2, 130]],
[66, [2, 132]],
[67, [2, 134]],
[68, [3, 204]],
[69, [6, 414]],
[70, [2, 140]],
[71, [2, 142]],
[72, [2, 144]],
[73, [14, 1022]],
[74, [3, 222]],
[75, [4, 300]],
[76, [4, 304]],
[77, [3, 231]],
[78, [3, 234]],
[79, [17, 1343]],
[80, [3, 240]],
[81, [3, 243]],
[82, [5, 410]],
[83, [4, 332]],
[84, [5, 420]],
[85, [4, 340]],
[86, [4, 344]],
[87, [12, 1044]],
[88, [5, 440]],
[89, [16, 1424]],
[90, [16, 1440]],
[91, [11, 1001]],
[92, [11, 1012]],
[93, [11, 1023]],
[94, [11, 1034]],
[95, [12, 1140]],
[96, [14, 1344]],
[97, [22, 2134]],
[98, [33, 3234]],
[99, [1156, 114444]],
[100, [1, 100]],
[101, [1, 101]],
[102, [1, 102]],
[103, [1, 103]],
[104, [1, 104]],
[105, [2, 210]],
[106, [2, 212]],
[107, [2, 214]],
[108, [3, 324]],
[109, [37, 4033]],
[110, [1, 110]],
[111, [1, 111]],
[112, [1, 112]],
[113, [1, 113]],
[114, [1, 114]],
[115, [2, 230]],
[116, [2, 232]],
[117, [2, 234]],
[118, [18, 2124]],
[119, [17, 2023]],
[120, [1, 120]],
[121, [1, 121]],
[122, [1, 122]],
[123, [1, 123]],
[124, [1, 124]],
[125, [8, 1000]],
[126, [9, 1134]],
[127, [9, 1143]],
[128, [8, 1024]],
[129, [8, 1032]],
[130, [1, 130]],
[131, [1, 131]],
[132, [1, 132]],
[133, [1, 133]],
[134, [1, 134]],
[135, [18, 2430]],
[136, [9, 1224]],
[137, [3, 411]],
[138, [3, 414]],
[139, [8, 1112]],
[140, [1, 140]],
[141, [1, 141]],
[142, [1, 142]],
[143, [1, 143]],
[144, [1, 144]],
[145, [14, 2030]],
[146, [7, 1022]],
[147, [3, 441]],
[148, [3, 444]],
[149, [7, 1043]],
[150, [2, 300]],
[151, [2, 302]],
[152, [2, 304]],
[153, [8, 1224]],
[154, [8, 1232]],
[155, [2, 310]],
[156, [2, 312]],
[157, [2, 314]],
[158, [9, 1422]],
[159, [7, 1113]],
[160, [2, 320]],
[161, [2, 322]],
[162, [2, 324]],
[163, [7, 1141]],
[164, [8, 1312]],
[165, [2, 330]],
[166, [2, 332]],
[167, [2, 334]],
[168, [8, 1344]],
[169, [6, 1014]],
[170, [2, 340]],
[171, [2, 342]],
[172, [2, 344]],
[173, [7, 1211]],
[174, [6, 1044]],
[175, [8, 1400]],
[176, [7, 1232]],
[177, [12, 2124]],
[178, [8, 1424]],
[179, [8, 1432]],
[180, [8, 1440]],
[181, [24, 4344]],
[182, [11, 2002]],
[183, [11, 2013]],
[184, [6, 1104]],
[185, [6, 1110]],
[186, [7, 1302]],
[187, [6, 1122]],
[188, [13, 2444]],
[189, [6, 1134]],
[190, [6, 1140]],
[191, [11, 2101]],
[192, [7, 1344]],
[193, [11, 2123]],
[194, [11, 2134]],
[195, [12, 2340]],
[196, [17, 3332]],
[197, [22, 4334]],
[198, [578, 114444]],
[199, [56, 11144]],
[200, [1, 200]],
[201, [1, 201]],
[202, [1, 202]],
[203, [1, 203]],
[204, [1, 204]],
[205, [2, 410]],
[206, [2, 412]],
[207, [2, 414]],
[208, [5, 1040]],
[209, [16, 3344]],
[210, [1, 210]],
[211, [1, 211]],
[212, [1, 212]],
[213, [1, 213]],
[214, [1, 214]],
[215, [2, 430]],
[216, [2, 432]],
[217, [2, 434]],
[218, [19, 4142]],
[219, [6, 1314]],
[220, [1, 220]],
[221, [1, 221]],
[222, [1, 222]],
[223, [1, 223]],
[224, [1, 224]],
[225, [64, 14400]],
[226, [5, 1130]],
[227, [9, 2043]],
[228, [5, 1140]],
[229, [18, 4122]],
[230, [1, 230]],
[231, [1, 231]],
[232, [1, 232]],
[233, [1, 233]],
[234, [1, 234]],
[235, [6, 1410]],
[236, [9, 2124]],
[237, [6, 1422]],
[238, [9, 2142]],
[239, [6, 1434]],
[240, [1, 240]],
[241, [1, 241]],
[242, [1, 242]],
[243, [1, 243]],
[244, [1, 244]],
[245, [14, 3430]],
[246, [5, 1230]],
[247, [9, 2223]],
[248, [5, 1240]],
[249, [9, 2241]],
[250, [4, 1000]],
[251, [4, 1004]],
[252, [12, 3024]],
[253, [4, 1012]],
[254, [8, 2032]],
[255, [4, 1020]],
[256, [4, 1024]],
[257, [9, 2313]],
[258, [4, 1032]],
[259, [9, 2331]],
[260, [4, 1040]],
[261, [4, 1044]],
[262, [5, 1310]],
[263, [8, 2104]],
[264, [5, 1320]],
[265, [8, 2120]],
[266, [5, 1330]],
[267, [9, 2403]],
[268, [5, 1340]],
[269, [9, 2421]],
[270, [9, 2430]],
[271, [41, 11111]],
[272, [42, 11424]],
[273, [11, 3003]],
[274, [11, 3014]],
[275, [4, 1100]],
[276, [4, 1104]],
[277, [12, 3324]],
[278, [4, 1112]],
[279, [8, 2232]],
[280, [4, 1120]],
[281, [4, 1124]],
[282, [5, 1410]],
[283, [4, 1132]],
[284, [5, 1420]],
[285, [4, 1140]],
[286, [4, 1144]],
[287, [12, 3444]],
[288, [5, 1440]],
[289, [7, 2023]],
[290, [7, 2030]],
[291, [11, 3201]],
[292, [7, 2044]],
[293, [8, 2344]],
[294, [11, 3234]],
[295, [14, 4130]],
[296, [14, 4144]],
[297, [419, 124443]],
[298, [34, 10132]],
[299, [67, 20033]],
[300, [1, 300]],
[301, [1, 301]],
[302, [1, 302]],
[303, [1, 303]],
[304, [1, 304]],
[305, [4, 1220]],
[306, [4, 1224]],
[307, [33, 10131]],
[308, [4, 1232]],
[309, [36, 11124]],
[310, [1, 310]],
[311, [1, 311]],
[312, [1, 312]],
[313, [1, 313]],
[314, [1, 314]],
[315, [14, 4410]],
[316, [7, 2212]],
[317, [13, 4121]],
[318, [13, 4134]],
[319, [7, 2233]],
[320, [1, 320]],
[321, [1, 321]],
[322, [1, 322]],
[323, [1, 323]],
[324, [1, 324]],
[325, [4, 1300]],
[326, [4, 1304]],
[327, [93, 30411]],
[328, [4, 1312]],
[329, [7, 2303]],
[330, [1, 330]],
[331, [1, 331]],
[332, [1, 332]],
[333, [1, 333]],
[334, [1, 334]],
[335, [4, 1340]],
[336, [4, 1344]],
[337, [3, 1011]],
[338, [3, 1014]],
[339, [6, 2034]],
[340, [1, 340]],
[341, [1, 341]],
[342, [1, 342]],
[343, [1, 343]],
[344, [1, 344]],
[345, [12, 4140]],
[346, [7, 2422]],
[347, [3, 1041]],
[348, [3, 1044]],
[349, [7, 2443]],
[350, [4, 1400]],
[351, [4, 1404]],
[352, [6, 2112]],
[353, [4, 1412]],
[354, [6, 2124]],
[355, [4, 1420]],
[356, [4, 1424]],
[357, [6, 2142]],
[358, [4, 1432]],
[359, [9, 3231]],
[360, [4, 1440]],
[361, [4, 1444]],
[362, [12, 4344]],
[363, [34, 12342]],
[364, [11, 4004]],
[365, [28, 10220]],
[366, [34, 12444]],
[367, [3, 1101]],
[368, [3, 1104]],
[369, [6, 2214]],
[370, [3, 1110]],
[371, [3, 1113]],
[372, [6, 2232]],
[373, [11, 4103]],
[374, [3, 1122]],
[375, [8, 3000]],
[376, [32, 12032]],
[377, [3, 1131]],
[378, [3, 1134]],
[379, [8, 3032]],
[380, [3, 1140]],
[381, [3, 1143]],
[382, [11, 4202]],
[383, [11, 4213]],
[384, [6, 2304]],
[385, [6, 2310]],
[386, [27, 10422]],
[387, [6, 2322]],
[388, [8, 3104]],
[389, [6, 2334]],
[390, [6, 2340]],
[391, [11, 4301]],
[392, [11, 4312]],
[393, [8, 3144]],
[394, [11, 4334]],
[395, [34, 13430]],
[396, [289, 114444]],
[397, [26, 10322]],
[398, [28, 11144]],
[399, [56, 22344]],
[400, [1, 400]],
[401, [1, 401]],
[402, [1, 402]],
[403, [1, 403]],
[404, [1, 404]],
[405, [6, 2430]],
[406, [5, 2030]],
[407, [3, 1221]],
[408, [3, 1224]],
[409, [27, 11043]],
[410, [1, 410]],
[411, [1, 411]],
[412, [1, 412]],
[413, [1, 413]],
[414, [1, 414]],
[415, [8, 3320]],
[416, [25, 10400]],
[417, [32, 13344]],
[418, [8, 3344]],
[419, [27, 11313]],
[420, [1, 420]],
[421, [1, 421]],
[422, [1, 422]],
[423, [1, 423]],
[424, [1, 424]],
[425, [8, 3400]],
[426, [5, 2130]],
[427, [26, 11102]],
[428, [5, 2140]],
[429, [7, 3003]],
[430, [1, 430]],
[431, [1, 431]],
[432, [1, 432]],
[433, [1, 433]],
[434, [1, 434]],
[435, [24, 10440]],
[436, [79, 34444]],
[437, [3, 1311]],
[438, [3, 1314]],
[439, [26, 11414]],
[440, [1, 440]],
[441, [1, 441]],
[442, [1, 442]],
[443, [1, 443]],
[444, [1, 444]],
[445, [32, 14240]],
[446, [5, 2230]],
[447, [3, 1341]],
[448, [3, 1344]],
[449, [7, 3143]],
[450, [32, 14400]],
[451, [32, 14432]],
[452, [25, 11300]],
[453, [27, 12231]],
[454, [23, 10442]],
[455, [22, 10010]],
[456, [9, 4104]],
[457, [9, 4113]],
[458, [9, 4122]],
[459, [7, 3213]],
[460, [5, 2300]],
[461, [22, 10142]],
[462, [5, 2310]],
[463, [7, 3241]],
[464, [5, 2320]],
[465, [22, 10230]],
[466, [5, 2330]],
[467, [3, 1401]],
[468, [3, 1404]],
[469, [9, 4221]],
[470, [3, 1410]],
[471, [3, 1413]],
[472, [7, 3304]],
[473, [7, 3311]],
[474, [3, 1422]],
[475, [24, 11400]],
[476, [7, 3332]],
[477, [3, 1431]],
[478, [3, 1434]],
[479, [9, 4311]],
[480, [3, 1440]],
[481, [3, 1443]],
[482, [5, 2410]],
[483, [21, 10143]],
[484, [5, 2420]],
[485, [44, 21340]],
[486, [5, 2430]],
[487, [23, 11201]],
[488, [5, 2440]],
[489, [7, 3423]],
[490, [7, 3430]],
[491, [21, 10311]],
[492, [7, 3444]],
[493, [27, 13311]],
[494, [43, 21242]],
[495, [2312, 1144440]],
[496, [25, 12400]],
[497, [23, 11431]],
[498, [29, 14442]],
[499, [67, 33433]],
[500, [2, 1000]],
[501, [2, 1002]],
[502, [2, 1004]],
[503, [4, 2012]],
[504, [6, 3024]],
[505, [2, 1010]],
[506, [2, 1012]],
[507, [2, 1014]],
[508, [4, 2032]],
[509, [26, 13234]],
[510, [2, 1020]],
[511, [2, 1022]],
[512, [2, 1024]],
[513, [8, 4104]],
[514, [8, 4112]],
[515, [2, 1030]],
[516, [2, 1032]],
[517, [2, 1034]],
[518, [8, 4144]],
[519, [6, 3114]],
[520, [2, 1040]],
[521, [2, 1042]],
[522, [2, 1044]],
[523, [27, 14121]],
[524, [6, 3144]],
[525, [4, 2100]],
[526, [4, 2104]],
[527, [19, 10013]],
[528, [4, 2112]],
[529, [8, 4232]],
[530, [4, 2120]],
[531, [4, 2124]],
[532, [25, 13300]],
[533, [4, 2132]],
[534, [6, 3204]],
[535, [4, 2140]],
[536, [4, 2144]],
[537, [6, 3222]],
[538, [8, 4304]],
[539, [6, 3234]],
[540, [6, 3240]],
[541, [23, 12443]],
[542, [41, 22222]],
[543, [8, 4344]],
[544, [21, 11424]],
[545, [74, 40330]],
[546, [22, 12012]],
[547, [22, 12034]],
[548, [19, 10412]],
[549, [19, 10431]],
[550, [2, 1100]],
[551, [2, 1102]],
[552, [2, 1104]],
[553, [4, 2212]],
[554, [6, 3324]],
[555, [2, 1110]],
[556, [2, 1112]],
[557, [2, 1114]],
[558, [4, 2232]],
[559, [36, 20124]],
[560, [2, 1120]],
[561, [2, 1122]],
[562, [2, 1124]],
[563, [18, 10134]],
[564, [25, 14100]],
[565, [2, 1130]],
[566, [2, 1132]],
[567, [2, 1134]],
[568, [18, 10224]],
[569, [6, 3414]],
[570, [2, 1140]],
[571, [2, 1142]],
[572, [2, 1144]],
[573, [7, 4011]],
[574, [6, 3444]],
[575, [4, 2300]],
[576, [4, 2304]],
[577, [42, 24234]],
[578, [4, 2312]],
[579, [18, 10422]],
[580, [4, 2320]],
[581, [4, 2324]],
[582, [21, 12222]],
[583, [4, 2332]],
[584, [23, 13432]],
[585, [4, 2340]],
[586, [4, 2344]],
[587, [36, 21132]],
[588, [24, 14112]],
[589, [7, 4123]],
[590, [7, 4130]],
[591, [21, 12411]],
[592, [7, 4144]],
[593, [24, 14232]],
[594, [226, 134244]],
[595, [34, 20230]],
[596, [17, 10132]],
[597, [19, 11343]],
[598, [34, 20332]],
[599, [57, 34143]],
[600, [2, 1200]],
[601, [2, 1202]],
[602, [2, 1204]],
[603, [4, 2412]],
[604, [5, 3020]],
[605, [2, 1210]],
[606, [2, 1212]],
[607, [2, 1214]],
[608, [4, 2432]],
[609, [38, 23142]],
[610, [2, 1220]],
[611, [2, 1222]],
[612, [2, 1224]],
[613, [17, 10421]],
[614, [23, 14122]],
[615, [2, 1230]],
[616, [2, 1232]],
[617, [2, 1234]],
[618, [18, 11124]],
[619, [7, 4333]],
[620, [2, 1240]],
[621, [2, 1242]],
[622, [2, 1244]],
[623, [18, 11214]],
[624, [5, 3120]],
[625, [16, 10000]],
[626, [5, 3130]],
[627, [16, 10032]],
[628, [5, 3140]],
[629, [7, 4403]],
[630, [7, 4410]],
[631, [53, 33443]],
[632, [7, 4424]],
[633, [7, 4431]],
[634, [16, 10144]],
[635, [18, 11430]],
[636, [54, 34344]],
[637, [19, 12103]],
[638, [19, 12122]],
[639, [16, 10224]],
[640, [5, 3200]],
[641, [22, 14102]],
[642, [5, 3210]],
[643, [38, 24434]],
[644, [5, 3220]],
[645, [16, 10320]],
[646, [5, 3230]],
[647, [22, 14234]],
[648, [5, 3240]],
[649, [17, 11033]],
[650, [2, 1300]],
[651, [2, 1302]],
[652, [2, 1304]],
[653, [17, 11101]],
[654, [63, 41202]],
[655, [2, 1310]],
[656, [2, 1312]],
[657, [2, 1314]],
[658, [35, 23030]],
[659, [17, 11203]],
[660, [2, 1320]],
[661, [2, 1322]],
[662, [2, 1324]],
[663, [61, 40443]],
[664, [5, 3320]],
[665, [2, 1330]],
[666, [2, 1332]],
[667, [2, 1334]],
[668, [3, 2004]],
[669, [6, 4014]],
[670, [2, 1340]],
[671, [2, 1342]],
[672, [2, 1344]],
[673, [17, 11441]],
[674, [3, 2022]],
[675, [36, 24300]],
[676, [15, 10140]],
[677, [3, 2031]],
[678, [3, 2034]],
[679, [18, 12222]],
[680, [3, 2040]],
[681, [3, 2043]],
[682, [5, 3410]],
[683, [21, 14343]],
[684, [5, 3420]],
[685, [6, 4110]],
[686, [5, 3430]],
[687, [6, 4122]],
[688, [5, 3440]],
[689, [6, 4134]],
[690, [6, 4140]],
[691, [31, 21421]],
[692, [32, 22144]],
[693, [194, 134442]],
[694, [15, 10410]],
[695, [16, 11120]],
[696, [15, 10440]],
[697, [19, 13243]],
[698, [29, 20242]],
[699, [59, 41241]],
[700, [2, 1400]],
[701, [2, 1402]],
[702, [2, 1404]],
[703, [47, 33041]],
[704, [3, 2112]],
[705, [2, 1410]],
[706, [2, 1412]],
[707, [2, 1414]],
[708, [3, 2124]],
[709, [16, 11344]],
[710, [2, 1420]],
[711, [2, 1422]],
[712, [2, 1424]],
[713, [17, 12121]],
[714, [3, 2142]],
[715, [2, 1430]],
[716, [2, 1432]],
[717, [2, 1434]],
[718, [28, 20104]],
[719, [6, 4314]],
[720, [2, 1440]],
[721, [2, 1442]],
[722, [2, 1444]],
[723, [14, 10122]],
[724, [6, 4344]],
[725, [28, 20300]],
[726, [17, 12342]],
[727, [46, 33442]],
[728, [18, 13104]],
[729, [18, 13122]],
[730, [14, 10220]],
[731, [14, 10234]],
[732, [17, 12444]],
[733, [137, 100421]],
[734, [3, 2202]],
[735, [6, 4410]],
[736, [14, 10304]],
[737, [3, 2211]],
[738, [3, 2214]],
[739, [6, 4434]],
[740, [3, 2220]],
[741, [3, 2223]],
[742, [15, 11130]],
[743, [14, 10402]],
[744, [3, 2232]],
[745, [14, 10430]],
[746, [14, 10444]],
[747, [3, 2241]],
[748, [3, 2244]],
[749, [19, 14231]],
[750, [4, 3000]],
[751, [4, 3004]],
[752, [16, 12032]],
[753, [4, 3012]],
[754, [15, 11310]],
[755, [4, 3020]],
[756, [4, 3024]],
[757, [16, 12112]],
[758, [4, 3032]],
[759, [16, 12144]],
[760, [4, 3040]],
[761, [4, 3044]],
[762, [15, 11430]],
[763, [54, 41202]],
[764, [16, 12224]],
[765, [16, 12240]],
[766, [17, 13022]],
[767, [3, 2301]],
[768, [3, 2304]],
[769, [16, 12304]],
[770, [3, 2310]],
[771, [3, 2313]],
[772, [17, 13124]],
[773, [17, 13141]],
[774, [3, 2322]],
[775, [4, 3100]],
[776, [4, 3104]],
[777, [3, 2331]],
[778, [3, 2334]],
[779, [17, 13243]],
[780, [3, 2340]],
[781, [3, 2343]],
[782, [26, 20332]],
[783, [4, 3132]],
[784, [18, 14112]],
[785, [4, 3140]],
[786, [4, 3144]],
[787, [13, 10231]],
[788, [13, 10244]],
[789, [17, 13413]],
[790, [17, 13430]],
[791, [41, 32431]],
[792, [157, 124344]],
[793, [14, 11102]],
[794, [13, 10322]],
[795, [14, 11130]],
[796, [14, 11144]],
[797, [29, 23113]],
[798, [28, 22344]],
[799, [167, 133433]],
[800, [3, 2400]],
[801, [3, 2403]],
[802, [5, 4010]],
[803, [4, 3212]],
[804, [3, 2412]],
[805, [4, 3220]],
[806, [4, 3224]],
[807, [3, 2421]],
[808, [3, 2424]],
[809, [26, 21034]],
[810, [3, 2430]],
[811, [3, 2433]],
[812, [25, 20300]],
[813, [41, 33333]],
[814, [3, 2442]],
[815, [14, 11410]],
[816, [14, 11424]],
[817, [26, 21242]],
[818, [124, 101432]],
[819, [16, 13104]],
[820, [5, 4100]],
[821, [54, 44334]],
[822, [5, 4110]],
[823, [27, 22221]],
[824, [5, 4120]],
[825, [4, 3300]],
[826, [4, 3304]],
[827, [16, 13232]],
[828, [4, 3312]],
[829, [28, 23212]],
[830, [4, 3320]],
[831, [4, 3324]],
[832, [16, 13312]],
[833, [4, 3332]],
[834, [16, 13344]],
[835, [4, 3340]],
[836, [4, 3344]],
[837, [12, 10044]],
[838, [24, 20112]],
[839, [16, 13424]],
[840, [5, 4200]],
[841, [123, 103443]],
[842, [5, 4210]],
[843, [17, 14331]],
[844, [5, 4220]],
[845, [12, 10140]],
[846, [5, 4230]],
[847, [13, 11011]],
[848, [5, 4240]],
[849, [17, 14433]],
[850, [4, 3400]],
[851, [4, 3404]],
[852, [12, 10224]],
[853, [4, 3412]],
[854, [13, 11102]],
[855, [4, 3420]],
[856, [4, 3424]],
[857, [13, 11141]],
[858, [4, 3432]],
[859, [26, 22334]],
[860, [4, 3440]],
[861, [4, 3444]],
[862, [5, 4310]],
[863, [27, 23301]],
[864, [5, 4320]],
[865, [14, 12110]],
[866, [5, 4330]],
[867, [12, 10404]],
[868, [5, 4340]],
[869, [28, 24332]],
[870, [12, 10440]],
[871, [13, 11323]],
[872, [46, 40112]],
[873, [14, 12222]],
[874, [15, 13110]],
[875, [16, 14000]],
[876, [15, 13140]],
[877, [13, 11401]],
[878, [13, 11414]],
[879, [38, 33402]],
[880, [5, 4400]],
[881, [14, 12334]],
[882, [5, 4410]],
[883, [34, 30022]],
[884, [5, 4420]],
[885, [24, 21240]],
[886, [5, 4430]],
[887, [23, 20401]],
[888, [5, 4440]],
[889, [16, 14224]],
[890, [16, 14240]],
[891, [162, 144342]],
[892, [25, 22300]],
[893, [24, 21432]],
[894, [15, 13410]],
[895, [16, 14320]],
[896, [15, 13440]],
[897, [26, 23322]],
[898, [35, 31430]],
[899, [156, 140244]],
[900, [16, 14400]],
[901, [111, 100011]],
[902, [16, 14432]],
[903, [37, 33411]],
[904, [111, 100344]],
[905, [48, 43440]],
[906, [124, 112344]],
[907, [49, 44443]],
[908, [115, 104420]],
[909, [1116, 1014444]],
[910, [11, 10010]],
[911, [11, 10021]],
[912, [11, 10032]],
[913, [11, 10043]],
[914, [23, 21022]],
[915, [22, 20130]],
[916, [34, 31144]],
[917, [12, 11004]],
[918, [23, 21114]],
[919, [37, 34003]],
[920, [11, 10120]],
[921, [11, 10131]],
[922, [11, 10142]],
[923, [48, 44304]],
[924, [13, 12012]],
[925, [12, 11100]],
[926, [12, 11112]],
[927, [12, 11124]],
[928, [23, 21344]],
[929, [36, 33444]],
[930, [11, 10230]],
[931, [11, 10241]],
[932, [25, 23300]],
[933, [131, 122223]],
[934, [13, 12142]],
[935, [12, 11220]],
[936, [12, 11232]],
[937, [12, 11244]],
[938, [14, 13132]],
[939, [26, 24414]],
[940, [11, 10340]],
[941, [13, 12233]],
[942, [12, 11304]],
[943, [14, 13202]],
[944, [35, 33040]],
[945, [12, 11340]],
[946, [14, 13244]],
[947, [13, 12311]],
[948, [13, 12324]],
[949, [109, 103441]],
[950, [12, 11400]],
[951, [12, 11412]],
[952, [12, 11424]],
[953, [14, 13342]],
[954, [13, 12402]],
[955, [22, 21010]],
[956, [15, 14340]],
[957, [13, 12441]],
[958, [14, 13412]],
[959, [46, 44114]],
[960, [14, 13440]],
[961, [22, 21142]],
[962, [15, 14430]],
[963, [21, 20223]],
[964, [21, 20244]],
[965, [22, 21230]],
[966, [105, 101430]],
[967, [23, 22241]],
[968, [24, 23232]],
[969, [116, 112404]],
[970, [22, 21340]],
[971, [23, 22333]],
[972, [21, 20412]],
[973, [21, 20433]],
[974, [23, 22402]],
[975, [24, 23400]],
[976, [24, 23424]],
[977, [33, 32241]],
[978, [35, 34230]],
[979, [127, 124333]],
[980, [33, 32340]],
[981, [31, 30411]],
[982, [31, 30442]],
[983, [34, 33422]],
[984, [35, 34440]],
[985, [44, 43340]],
[986, [42, 41412]],
[987, [43, 42441]],
[988, [133, 131404]],
[989, [236, 233404]],
[990, [1156, 1144440]],
[991, [111, 110001]],
[992, [111, 110112]],
[993, [111, 110223]],
[994, [111, 110334]],
[995, [112, 111440]],
[996, [142, 141432]],
[997, [222, 221334]],
[998, [333, 332334]],
[999, [111556, 111444444]],
[1000, [1, 1000]]]
And the smallest k
, such that n*k
contains only even digits:
[[1, [2, 2]],
[2, [1, 2]],
[3, [2, 6]],
[4, [1, 4]],
[5, [4, 20]],
[6, [1, 6]],
[7, [4, 28]],
[8, [1, 8]],
[9, [32, 288]],
[10, [2, 20]],
[11, [2, 22]],
[12, [2, 24]],
[13, [2, 26]],
[14, [2, 28]],
[15, [4, 60]],
[16, [3, 48]],
[17, [4, 68]],
[18, [16, 288]],
[19, [12, 228]],
[20, [1, 20]],
[21, [2, 42]],
[22, [1, 22]],
[23, [2, 46]],
[24, [1, 24]],
[25, [8, 200]],
[26, [1, 26]],
[27, [18, 486]],
[28, [1, 28]],
[29, [14, 406]],
[30, [2, 60]],
[31, [2, 62]],
[32, [2, 64]],
[33, [2, 66]],
[34, [2, 68]],
[35, [8, 280]],
[36, [8, 288]],
[37, [6, 222]],
[38, [6, 228]],
[39, [12, 468]],
[40, [1, 40]],
[41, [2, 82]],
[42, [1, 42]],
[43, [2, 86]],
[44, [1, 44]],
[45, [64, 2880]],
[46, [1, 46]],
[47, [6, 282]],
[48, [1, 48]],
[49, [14, 686]],
[50, [4, 200]],
[51, [4, 204]],
[52, [4, 208]],
[53, [8, 424]],
[54, [9, 486]],
[55, [4, 220]],
[56, [4, 224]],
[57, [4, 228]],
[58, [7, 406]],
[59, [14, 826]],
[60, [1, 60]],
[61, [4, 244]],
[62, [1, 62]],
[63, [14, 882]],
[64, [1, 64]],
[65, [4, 260]],
[66, [1, 66]],
[67, [4, 268]],
[68, [1, 68]],
[69, [12, 828]],
[70, [4, 280]],
[71, [4, 284]],
[72, [4, 288]],
[73, [28, 2044]],
[74, [3, 222]],
[75, [8, 600]],
[76, [3, 228]],
[77, [6, 462]],
[78, [6, 468]],
[79, [34, 2686]],
[80, [1, 80]],
[81, [6, 486]],
[82, [1, 82]],
[83, [8, 664]],
[84, [1, 84]],
[85, [8, 680]],
[86, [1, 86]],
[87, [24, 2088]],
[88, [1, 88]],
[89, [32, 2848]],
[90, [32, 2880]],
[91, [22, 2002]],
[92, [5, 460]],
[93, [22, 2046]],
[94, [3, 282]],
[95, [24, 2280]],
[96, [3, 288]],
[97, [44, 4268]],
[98, [7, 686]],
[99, [2312, 228888]],
[100, [2, 200]],
[101, [2, 202]],
[102, [2, 204]],
[103, [2, 206]],
[104, [2, 208]],
[105, [4, 420]],
[106, [4, 424]],
[107, [4, 428]],
[108, [6, 648]],
[109, [74, 8066]],
[110, [2, 220]],
[111, [2, 222]],
[112, [2, 224]],
[113, [2, 226]],
[114, [2, 228]],
[115, [4, 460]],
[116, [4, 464]],
[117, [4, 468]],
[118, [7, 826]],
[119, [34, 4046]],
[120, [2, 240]],
[121, [2, 242]],
[122, [2, 244]],
[123, [2, 246]],
[124, [2, 248]],
[125, [16, 2000]],
[126, [7, 882]],
[127, [18, 2286]],
[128, [5, 640]],
[129, [16, 2064]],
[130, [2, 260]],
[131, [2, 262]],
[132, [2, 264]],
[133, [2, 266]],
[134, [2, 268]],
[135, [36, 4860]],
[136, [3, 408]],
[137, [6, 822]],
[138, [6, 828]],
[139, [16, 2224]],
[140, [2, 280]],
[141, [2, 282]],
[142, [2, 284]],
[143, [2, 286]],
[144, [2, 288]],
[145, [28, 4060]],
[146, [14, 2044]],
[147, [6, 882]],
[148, [3, 444]],
[149, [14, 2086]],
[150, [4, 600]],
[151, [4, 604]],
[152, [4, 608]],
[153, [16, 2448]],
[154, [3, 462]],
[155, [4, 620]],
[156, [3, 468]],
[157, [4, 628]],
[158, [17, 2686]],
[159, [14, 2226]],
[160, [3, 480]],
[161, [4, 644]],
[162, [3, 486]],
[163, [14, 2282]],
[164, [5, 820]],
[165, [4, 660]],
[166, [4, 664]],
[167, [4, 668]],
[168, [5, 840]],
[169, [12, 2028]],
[170, [4, 680]],
[171, [4, 684]],
[172, [4, 688]],
[173, [14, 2422]],
[174, [12, 2088]],
[175, [16, 2800]],
[176, [5, 880]],
[177, [24, 4248]],
[178, [16, 2848]],
[179, [16, 2864]],
[180, [16, 2880]],
[181, [48, 8688]],
[182, [11, 2002]],
[183, [22, 4026]],
[184, [11, 2024]],
[185, [12, 2220]],
[186, [11, 2046]],
[187, [12, 2244]],
[188, [11, 2068]],
[189, [12, 2268]],
[190, [12, 2280]],
[191, [22, 4202]],
[192, [14, 2688]],
[193, [22, 4246]],
[194, [22, 4268]],
[195, [24, 4680]],
[196, [33, 6468]],
[197, [44, 8668]],
[198, [1156, 228888]],
[199, [112, 22288]],
[200, [1, 200]],
[201, [2, 402]],
[202, [1, 202]],
[203, [2, 406]],
[204, [1, 204]],
[205, [4, 820]],
[206, [1, 206]],
[207, [4, 828]],
[208, [1, 208]],
[209, [32, 6688]],
[210, [2, 420]],
[211, [2, 422]],
[212, [2, 424]],
[213, [2, 426]],
[214, [2, 428]],
[215, [4, 860]],
[216, [3, 648]],
[217, [4, 868]],
[218, [37, 8066]],
[219, [12, 2628]],
[220, [1, 220]],
[221, [2, 442]],
[222, [1, 222]],
[223, [2, 446]],
[224, [1, 224]],
[225, [128, 28800]],
[226, [1, 226]],
[227, [18, 4086]],
[228, [1, 228]],
[229, [36, 8244]],
[230, [2, 460]],
[231, [2, 462]],
[232, [2, 464]],
[233, [2, 466]],
[234, [2, 468]],
[235, [12, 2820]],
[236, [18, 4248]],
[237, [12, 2844]],
[238, [17, 4046]],
[239, [12, 2868]],
[240, [1, 240]],
[241, [2, 482]],
[242, [1, 242]],
[243, [2, 486]],
[244, [1, 244]],
[245, [28, 6860]],
[246, [1, 246]],
[247, [18, 4446]],
[248, [1, 248]],
[249, [18, 4482]],
[250, [8, 2000]],
[251, [8, 2008]],
[252, [9, 2268]],
[253, [8, 2024]],
[254, [9, 2286]],
[255, [8, 2040]],
[256, [8, 2048]],
[257, [18, 4626]],
[258, [8, 2064]],
[259, [18, 4662]],
[260, [1, 260]],
[261, [8, 2088]],
[262, [1, 262]],
[263, [16, 4208]],
[264, [1, 264]],
[265, [16, 4240]],
[266, [1, 266]],
[267, [18, 4806]],
[268, [1, 268]],
[269, [18, 4842]],
[270, [18, 4860]],
[271, [82, 22222]],
[272, [9, 2448]],
[273, [22, 6006]],
[274, [3, 822]],
[275, [8, 2200]],
[276, [3, 828]],
[277, [24, 6648]],
[278, [8, 2224]],
[279, [16, 4464]],
[280, [1, 280]],
[281, [8, 2248]],
[282, [1, 282]],
[283, [8, 2264]],
[284, [1, 284]],
[285, [8, 2280]],
[286, [1, 286]],
[287, [24, 6888]],
[288, [1, 288]],
[289, [14, 4046]],
[290, [14, 4060]],
[291, [22, 6402]],
[292, [7, 2044]],
[293, [16, 4688]],
[294, [3, 882]],
[295, [28, 8260]],
[296, [3, 888]],
[297, [838, 248886]],
[298, [7, 2086]],
[299, [134, 40066]],
[300, [2, 600]],
[301, [2, 602]],
[302, [2, 604]],
[303, [2, 606]],
[304, [2, 608]],
[305, [8, 2440]],
[306, [8, 2448]],
[307, [66, 20262]],
[308, [8, 2464]],
[309, [72, 22248]],
[310, [2, 620]],
[311, [2, 622]],
[312, [2, 624]],
[313, [2, 626]],
[314, [2, 628]],
[315, [28, 8820]],
[316, [9, 2844]],
[317, [26, 8242]],
[318, [7, 2226]],
[319, [14, 4466]],
[320, [2, 640]],
[321, [2, 642]],
[322, [2, 644]],
[323, [2, 646]],
[324, [2, 648]],
[325, [8, 2600]],
[326, [7, 2282]],
[327, [186, 60822]],
[328, [8, 2624]],
[329, [14, 4606]],
[330, [2, 660]],
[331, [2, 662]],
[332, [2, 664]],
[333, [2, 666]],
[334, [2, 668]],
[335, [8, 2680]],
[336, [8, 2688]],
[337, [6, 2022]],
[338, [6, 2028]],
[339, [12, 4068]],
[340, [2, 680]],
[341, [2, 682]],
[342, [2, 684]],
[343, [2, 686]],
[344, [2, 688]],
[345, [24, 8280]],
[346, [7, 2422]],
[347, [6, 2082]],
[348, [6, 2088]],
[349, [14, 4886]],
[350, [8, 2800]],
[351, [8, 2808]],
[352, [7, 2464]],
[353, [8, 2824]],
[354, [12, 4248]],
[355, [8, 2840]],
[356, [8, 2848]],
[357, [12, 4284]],
[358, [8, 2864]],
[359, [18, 6462]],
[360, [8, 2880]],
[361, [8, 2888]],
[362, [24, 8688]],
[363, [68, 24684]],
[364, [11, 4004]],
[365, [56, 20440]],
[366, [11, 4026]],
[367, [6, 2202]],
[368, [6, 2208]],
[369, [12, 4428]],
[370, [6, 2220]],
[371, [6, 2226]],
[372, [7, 2604]],
[373, [22, 8206]],
[374, [6, 2244]],
[375, [16, 6000]],
[376, [13, 4888]],
[377, [6, 2262]],
[378, [6, 2268]],
[379, [16, 6064]],
[380, [6, 2280]],
[381, [6, 2286]],
[382, [11, 4202]],
[383, [22, 8426]],
[384, [7, 2688]],
[385, [12, 4620]],
[386, [11, 4246]],
[387, [12, 4644]],
[388, [11, 4268]],
[389, [12, 4668]],
[390, [12, 4680]],
[391, [22, 8602]],
[392, [17, 6664]],
[393, [16, 6288]],
[394, [22, 8668]],
[395, [68, 26860]],
[396, [578, 228888]],
[397, [52, 20644]],
[398, [56, 22288]],
[399, [112, 44688]],
[400, [1, 400]],
[401, [2, 802]],
[402, [1, 402]],
[403, [2, 806]],
[404, [1, 404]],
[405, [12, 4860]],
[406, [1, 406]],
[407, [6, 2442]],
[408, [1, 408]],
[409, [54, 22086]],
[410, [2, 820]],
[411, [2, 822]],
[412, [2, 824]],
[413, [2, 826]],
[414, [2, 828]],
[415, [16, 6640]],
[416, [5, 2080]],
[417, [64, 26688]],
[418, [16, 6688]],
[419, [54, 22626]],
[420, [1, 420]],
[421, [2, 842]],
[422, [1, 422]],
[423, [2, 846]],
[424, [1, 424]],
[425, [16, 6800]],
[426, [1, 426]],
[427, [52, 22204]],
[428, [1, 428]],
[429, [14, 6006]],
[430, [2, 860]],
[431, [2, 862]],
[432, [2, 864]],
[433, [2, 866]],
[434, [2, 868]],
[435, [48, 20880]],
[436, [19, 8284]],
[437, [6, 2622]],
[438, [6, 2628]],
[439, [52, 22828]],
[440, [1, 440]],
[441, [2, 882]],
[442, [1, 442]],
[443, [2, 886]],
[444, [1, 444]],
[445, [64, 28480]],
[446, [1, 446]],
[447, [6, 2682]],
[448, [1, 448]],
[449, [14, 6286]],
[450, [64, 28800]],
[451, [64, 28864]],
[452, [5, 2260]],
[453, [54, 24462]],
[454, [9, 4086]],
[455, [44, 20020]],
[456, [5, 2280]],
[457, [18, 8226]],
[458, [18, 8244]],
[459, [14, 6426]],
[460, [1, 460]],
[461, [44, 20284]],
[462, [1, 462]],
[463, [14, 6482]],
[464, [1, 464]],
[465, [44, 20460]],
[466, [1, 466]],
[467, [6, 2802]],
[468, [1, 468]],
[469, [18, 8442]],
[470, [6, 2820]],
[471, [6, 2826]],
[472, [9, 4248]],
[473, [14, 6622]],
[474, [6, 2844]],
[475, [48, 22800]],
[476, [9, 4284]],
[477, [6, 2862]],
[478, [6, 2868]],
[479, [18, 8622]],
[480, [1, 480]],
[481, [6, 2886]],
[482, [1, 482]],
[483, [42, 20286]],
[484, [1, 484]],
[485, [88, 42680]],
[486, [1, 486]],
[487, [46, 22402]],
[488, [1, 488]],
[489, [14, 6846]],
[490, [14, 6860]],
[491, [42, 20622]],
[492, [5, 2460]],
[493, [54, 26622]],
[494, [9, 4446]],
[495, [4624, 2288880]],
[496, [5, 2480]],
[497, [46, 22862]],
[498, [9, 4482]],
[499, [134, 66866]],
[500, [4, 2000]],
[501, [4, 2004]],
[502, [4, 2008]],
[503, [8, 4024]],
[504, [12, 6048]],
[505, [4, 2020]],
[506, [4, 2024]],
[507, [4, 2028]],
[508, [8, 4064]],
[509, [52, 26468]],
[510, [4, 2040]],
[511, [4, 2044]],
[512, [4, 2048]],
[513, [16, 8208]],
[514, [9, 4626]],
[515, [4, 2060]],
[516, [4, 2064]],
[517, [4, 2068]],
[518, [9, 4662]],
[519, [12, 6228]],
[520, [4, 2080]],
[521, [4, 2084]],
[522, [4, 2088]],
[523, [54, 28242]],
[524, [5, 2620]],
[525, [8, 4200]],
[526, [8, 4208]],
[527, [38, 20026]],
[528, [5, 2640]],
[529, [16, 8464]],
[530, [8, 4240]],
[531, [8, 4248]],
[532, [5, 2660]],
[533, [8, 4264]],
[534, [9, 4806]],
[535, [8, 4280]],
[536, [5, 2680]],
[537, [12, 6444]],
[538, [9, 4842]],
[539, [12, 6468]],
[540, [9, 4860]],
[541, [46, 24886]],
[542, [41, 22222]],
[543, [16, 8688]],
[544, [42, 22848]],
[545, [148, 80660]],
[546, [11, 6006]],
[547, [44, 24068]],
[548, [11, 6028]],
[549, [38, 20862]],
[550, [4, 2200]],
[551, [4, 2204]],
[552, [4, 2208]],
[553, [8, 4424]],
[554, [12, 6648]],
[555, [4, 2220]],
[556, [4, 2224]],
[557, [4, 2228]],
[558, [8, 4464]],
[559, [72, 40248]],
[560, [4, 2240]],
[561, [4, 2244]],
[562, [4, 2248]],
[563, [36, 20268]],
[564, [5, 2820]],
[565, [4, 2260]],
[566, [4, 2264]],
[567, [4, 2268]],
[568, [5, 2840]],
[569, [12, 6828]],
[570, [4, 2280]],
[571, [4, 2284]],
[572, [4, 2288]],
[573, [14, 8022]],
[574, [12, 6888]],
[575, [8, 4600]],
[576, [5, 2880]],
[577, [84, 48468]],
[578, [7, 4046]],
[579, [36, 20844]],
[580, [7, 4060]],
[581, [8, 4648]],
[582, [11, 6402]],
[583, [8, 4664]],
[584, [7, 4088]],
[585, [8, 4680]],
[586, [8, 4688]],
[587, [72, 42264]],
[588, [11, 6468]],
[589, [14, 8246]],
[590, [14, 8260]],
[591, [42, 24822]],
[592, [14, 8288]],
[593, [48, 28464]],
[594, [419, 248886]],
[595, [68, 40460]],
[596, [34, 20264]],
[597, [38, 22686]],
[598, [67, 40066]],
[599, [114, 68286]],
[600, [1, 600]],
[601, [4, 2404]],
[602, [1, 602]],
[603, [8, 4824]],
[604, [1, 604]],
[605, [4, 2420]],
[606, [1, 606]],
[607, [4, 2428]],
[608, [1, 608]],
[609, [76, 46284]],
[610, [4, 2440]],
[611, [4, 2444]],
[612, [4, 2448]],
[613, [34, 20842]],
[614, [33, 20262]],
[615, [4, 2460]],
[616, [4, 2464]],
[617, [4, 2468]],
[618, [36, 22248]],
[619, [14, 8666]],
[620, [1, 620]],
[621, [4, 2484]],
[622, [1, 622]],
[623, [36, 22428]],
[624, [1, 624]],
[625, [32, 20000]],
[626, [1, 626]],
[627, [32, 20064]],
[628, [1, 628]],
[629, [14, 8806]],
[630, [14, 8820]],
[631, [106, 66886]],
[632, [7, 4424]],
[633, [14, 8862]],
[634, [13, 8242]],
[635, [36, 22860]],
[636, [13, 8268]],
[637, [38, 24206]],
[638, [7, 4466]],
[639, [32, 20448]],
[640, [1, 640]],
[641, [44, 28204]],
[642, [1, 642]],
[643, [76, 48868]],
[644, [1, 644]],
[645, [32, 20640]],
[646, [1, 646]],
[647, [44, 28468]],
[648, [1, 648]],
[649, [34, 22066]],
[650, [4, 2600]],
[651, [4, 2604]],
[652, [4, 2608]],
[653, [34, 22202]],
[654, [93, 60822]],
[655, [4, 2620]],
[656, [4, 2624]],
[657, [4, 2628]],
[658, [7, 4606]],
[659, [34, 22406]],
[660, [1, 660]],
[661, [4, 2644]],
[662, [1, 662]],
[663, [122, 80886]],
[664, [1, 664]],
[665, [4, 2660]],
[666, [1, 666]],
[667, [4, 2668]],
[668, [1, 668]],
[669, [12, 8028]],
[670, [4, 2680]],
[671, [4, 2684]],
[672, [4, 2688]],
[673, [34, 22882]],
[674, [3, 2022]],
[675, [72, 48600]],
[676, [3, 2028]],
[677, [6, 4062]],
[678, [6, 4068]],
[679, [36, 24444]],
[680, [1, 680]],
[681, [6, 4086]],
[682, [1, 682]],
[683, [42, 28686]],
[684, [1, 684]],
[685, [12, 8220]],
[686, [1, 686]],
[687, [12, 8244]],
[688, [1, 688]],
[689, [12, 8268]],
[690, [12, 8280]],
[691, [62, 42842]],
[692, [7, 4844]],
[693, [388, 268884]],
[694, [3, 2082]],
[695, [32, 22240]],
[696, [3, 2088]],
[697, [38, 26486]],
[698, [7, 4886]],
[699, [118, 82482]],
[700, [4, 2800]],
[701, [4, 2804]],
[702, [4, 2808]],
[703, [94, 66082]],
[704, [6, 4224]],
[705, [4, 2820]],
[706, [4, 2824]],
[707, [4, 2828]],
[708, [6, 4248]],
[709, [32, 22688]],
[710, [4, 2840]],
[711, [4, 2844]],
[712, [4, 2848]],
[713, [34, 24242]],
[714, [6, 4284]],
[715, [4, 2860]],
[716, [4, 2864]],
[717, [4, 2868]],
[718, [9, 6462]],
[719, [12, 8628]],
[720, [4, 2880]],
[721, [4, 2884]],
[722, [4, 2888]],
[723, [28, 20244]],
[724, [12, 8688]],
[725, [56, 40600]],
[726, [34, 24684]],
[727, [92, 66884]],
[728, [11, 8008]],
[729, [36, 26244]],
[730, [28, 20440]],
[731, [28, 20468]],
[732, [34, 24888]],
[733, [274, 200842]],
[734, [3, 2202]],
[735, [12, 8820]],
[736, [3, 2208]],
[737, [6, 4422]],
[738, [6, 4428]],
[739, [12, 8868]],
[740, [3, 2220]],
[741, [6, 4446]],
[742, [3, 2226]],
[743, [28, 20804]],
[744, [6, 4464]],
[745, [28, 20860]],
[746, [11, 8206]],
[747, [6, 4482]],
[748, [3, 2244]],
[749, [38, 28462]],
[750, [8, 6000]],
[751, [8, 6008]],
[752, [32, 24064]],
[753, [8, 6024]],
[754, [3, 2262]],
[755, [8, 6040]],
[756, [3, 2268]],
[757, [32, 24224]],
[758, [8, 6064]],
[759, [32, 24288]],
[760, [3, 2280]],
[761, [8, 6088]],
[762, [3, 2286]],
[763, [108, 82404]],
[764, [11, 8404]],
[765, [32, 24480]],
[766, [11, 8426]],
[767, [6, 4602]],
[768, [6, 4608]],
[769, [32, 24608]],
[770, [6, 4620]],
[771, [6, 4626]],
[772, [27, 20844]],
[773, [34, 26282]],
[774, [6, 4644]],
[775, [8, 6200]],
[776, [8, 6208]],
[777, [6, 4662]],
[778, [6, 4668]],
[779, [34, 26486]],
[780, [6, 4680]],
[781, [6, 4686]],
[782, [11, 8602]],
[783, [8, 6264]],
[784, [11, 8624]],
[785, [8, 6280]],
[786, [8, 6288]],
[787, [26, 20462]],
[788, [11, 8668]],
[789, [34, 26826]],
[790, [34, 26860]],
[791, [82, 64862]],
[792, [289, 228888]],
[793, [28, 22204]],
[794, [26, 20644]],
[795, [28, 22260]],
[796, [28, 22288]],
[797, [58, 46226]],
[798, [56, 44688]],
[799, [334, 266866]],
[800, [1, 800]],
[801, [6, 4806]],
[802, [1, 802]],
[803, [8, 6424]],
[804, [1, 804]],
[805, [8, 6440]],
[806, [1, 806]],
[807, [6, 4842]],
[808, [1, 808]],
[809, [52, 42068]],
[810, [6, 4860]],
[811, [6, 4866]],
[812, [5, 4060]],
[813, [82, 66666]],
[814, [3, 2442]],
[815, [28, 22820]],
[816, [3, 2448]],
[817, [52, 42484]],
[818, [27, 22086]],
[819, [32, 26208]],
[820, [1, 820]],
[821, [108, 88668]],
[822, [1, 822]],
[823, [54, 44442]],
[824, [1, 824]],
[825, [8, 6600]],
[826, [1, 826]],
[827, [32, 26464]],
[828, [1, 828]],
[829, [56, 46424]],
[830, [8, 6640]],
[831, [8, 6648]],
[832, [25, 20800]],
[833, [8, 6664]],
[834, [32, 26688]],
[835, [8, 6680]],
[836, [8, 6688]],
[837, [24, 20088]],
[838, [27, 22626]],
[839, [32, 26848]],
[840, [1, 840]],
[841, [246, 206886]],
[842, [1, 842]],
[843, [34, 28662]],
[844, [1, 844]],
[845, [24, 20280]],
[846, [1, 846]],
[847, [26, 22022]],
[848, [1, 848]],
[849, [34, 28866]],
[850, [8, 6800]],
[851, [8, 6808]],
[852, [5, 4260]],
[853, [8, 6824]],
[854, [26, 22204]],
[855, [8, 6840]],
[856, [5, 4280]],
[857, [26, 22282]],
[858, [7, 6006]],
[859, [52, 44668]],
[860, [1, 860]],
[861, [8, 6888]],
[862, [1, 862]],
[863, [54, 46602]],
[864, [1, 864]],
[865, [28, 24220]],
[866, [1, 866]],
[867, [24, 20808]],
[868, [1, 868]],
[869, [56, 48664]],
[870, [24, 20880]],
[871, [26, 22646]],
[872, [79, 68888]],
[873, [28, 24444]],
[874, [3, 2622]],
[875, [32, 28000]],
[876, [3, 2628]],
[877, [26, 22802]],
[878, [26, 22828]],
[879, [76, 66804]],
[880, [1, 880]],
[881, [28, 24668]],
[882, [1, 882]],
[883, [68, 60044]],
[884, [1, 884]],
[885, [48, 42480]],
[886, [1, 886]],
[887, [46, 40802]],
[888, [1, 888]],
[889, [32, 28448]],
[890, [32, 28480]],
[891, [324, 288684]],
[892, [5, 4460]],
[893, [48, 42864]],
[894, [3, 2682]],
[895, [32, 28640]],
[896, [3, 2688]],
[897, [52, 46644]],
[898, [7, 6286]],
[899, [312, 280488]],
[900, [32, 28800]],
[901, [222, 200022]],
[902, [32, 28864]],
[903, [74, 66822]],
[904, [25, 22600]],
[905, [96, 86880]],
[906, [27, 24462]],
[907, [98, 88886]],
[908, [23, 20884]],
[909, [2232, 2028888]],
[910, [22, 20020]],
[911, [22, 20042]],
[912, [9, 8208]],
[913, [22, 20086]],
[914, [9, 8226]],
[915, [44, 40260]],
[916, [9, 8244]],
[917, [24, 22008]],
[918, [7, 6426]],
[919, [74, 68006]],
[920, [5, 4600]],
[921, [22, 20262]],
[922, [22, 20284]],
[923, [96, 88608]],
[924, [5, 4620]],
[925, [24, 22200]],
[926, [7, 6482]],
[927, [24, 22248]],
[928, [5, 4640]],
[929, [72, 66888]],
[930, [22, 20460]],
[931, [22, 20482]],
[932, [5, 4660]],
[933, [262, 244446]],
[934, [3, 2802]],
[935, [24, 22440]],
[936, [3, 2808]],
[937, [24, 22488]],
[938, [9, 8442]],
[939, [52, 48828]],
[940, [3, 2820]],
[941, [26, 24466]],
[942, [3, 2826]],
[943, [28, 26404]],
[944, [7, 6608]],
[945, [24, 22680]],
[946, [7, 6622]],
[947, [26, 24622]],
[948, [3, 2844]],
[949, [218, 206882]],
[950, [24, 22800]],
[951, [24, 22824]],
[952, [7, 6664]],
[953, [28, 26684]],
[954, [3, 2862]],
[955, [44, 42020]],
[956, [3, 2868]],
[957, [26, 24882]],
[958, [9, 8622]],
[959, [92, 88228]],
[960, [3, 2880]],
[961, [44, 42284]],
[962, [3, 2886]],
[963, [42, 40446]],
[964, [5, 4820]],
[965, [44, 42460]],
[966, [21, 20286]],
[967, [46, 44482]],
[968, [5, 4840]],
[969, [232, 224808]],
[970, [44, 42680]],
[971, [46, 44666]],
[972, [5, 4860]],
[973, [42, 40866]],
[974, [23, 22402]],
[975, [48, 46800]],
[976, [5, 4880]],
[977, [66, 64482]],
[978, [7, 6846]],
[979, [254, 248666]],
[980, [7, 6860]],
[981, [62, 60822]],
[982, [21, 20622]],
[983, [68, 66844]],
[984, [7, 6888]],
[985, [88, 86680]],
[986, [27, 26622]],
[987, [86, 84882]],
[988, [43, 42484]],
[989, [472, 466808]],
[990, [2312, 2288880]],
[991, [222, 220002]],
[992, [25, 24800]],
[993, [222, 220446]],
[994, [23, 22862]],
[995, [224, 222880]],
[996, [29, 28884]],
[997, [444, 442668]],
[998, [67, 66866]],
[999, [223112, 222888888]],
[1000, [2, 2000]]]
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The same sort of algorithm (the brute force one shown above) can be used to find other multipliers. For instance, here are the first 1000 numbers, and the smallest integer whose product includes only the digits
0
,6
and9
: