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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 |
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
and 9
:
[1, [6, 6]],
[2, [3, 6]],
[3, [2, 6]],
[4, [15, 60]],
[5, [12, 60]],
[6, [1, 6]],
[7, [87, 609]],
[8, [12, 96]],
[9, [1, 9]],
[10, [6, 60]],
[11, [6, 66]],
[12, [5, 60]],
[13, [462, 6006]],
[14, [69, 966]],
[15, [4, 60]],
[16, [6, 96]],
[17, [57, 969]],
[18, [5, 90]],
[19, [51, 969]],
[20, [3, 60]],
[21, [29, 609]],
[22, [3, 66]],
[23, [3, 69]],
[24, [4, 96]],
[25, [24, 600]],
[26, [231, 6006]],
[27, [37, 999]],
[28, [345, 9660]],
[29, [21, 609]],
[30, [2, 60]],
[31, [216, 6696]],
[32, [3, 96]],
[33, [2, 66]],
[34, [285, 9690]],
[35, [174, 6090]],
[36, [25, 900]],
[37, [18, 666]],
[38, [237, 9006]],
[39, [154, 6006]],
[40, [15, 600]],
[41, [1626, 66666]],
[42, [23, 966]],
[43, [162, 6966]],
[44, [15, 660]],
[45, [2, 90]],
[46, [15, 690]],
[47, [147, 6909]],
[48, [2, 96]],
[49, [141, 6909]],
[50, [12, 600]],
[51, [19, 969]],
[52, [1155, 60060]],
[53, [132, 6996]],
[54, [124, 6696]],
[55, [12, 660]],
[56, [1191, 66696]],
[57, [17, 969]],
[58, [12, 696]],
[59, [1134, 66906]],
[60, [1, 60]],
[61, [159, 9699]],
[62, [108, 6696]],
[63, [143, 9009]],
[64, [15, 960]],
[65, [924, 60060]],
[66, [1, 66]],
[67, [897, 60099]],
[68, [147, 9996]],
[69, [1, 69]],
[70, [87, 6090]],
[71, [846, 60066]],
[72, [93, 6696]],
[73, [822, 60006]],
[74, [9, 666]],
[75, [8, 600]],
[76, [921, 69996]],
[77, [78, 6006]],
[78, [77, 6006]],
[79, [114, 9006]],
[80, [12, 960]],
[81, [86, 6966]],
[82, [813, 66666]],
[83, [12, 996]],
[84, [115, 9660]],
[85, [114, 9690]],
[86, [81, 6966]],
[87, [7, 609]],
[88, [75, 6600]],
[89, [681, 60609]],
[90, [1, 90]],
[91, [66, 6006]],
[92, [75, 6900]],
[93, [72, 6696]],
[94, [639, 60066]],
[95, [102, 9690]],
[96, [1, 96]],
[97, [6867, 666099]],
[98, [102, 9996]],
[99, [1, 99]],
[100, [6, 600]],
[101, [6, 606]],
[102, [95, 9690]],
[103, [933, 96099]],
[104, [924, 96096]],
[105, [58, 6090]],
[106, [66, 6996]],
[107, [57, 6099]],
[108, [62, 6696]],
[109, [834, 90906]],
[110, [6, 660]],
[111, [6, 666]],
[112, [858, 96096]],
[113, [882, 99666]],
[114, [79, 9006]],
[115, [6, 690]],
[116, [6, 696]],
[117, [57, 6669]],
[118, [567, 66906]],
[119, [51, 6069]],
[120, [5, 600]],
[121, [546, 66066]],
[122, [573, 69906]],
[123, [542, 66666]],
[124, [54, 6696]],
[125, [48, 6000]],
[126, [481, 60606]],
[127, [48, 6096]],
[128, [75, 9600]],
[129, [54, 6966]],
[130, [462, 60060]],
[131, [4626, 606006]],
[132, [5, 660]],
[133, [4512, 600096]],
[134, [744, 99696]],
[135, [74, 9990]],
[136, [486, 66096]],
[137, [438, 60006]],
[138, [5, 690]],
[139, [654, 90906]],
[140, [69, 9660]],
[141, [49, 6909]],
[142, [423, 60066]],
[143, [42, 6006]],
[144, [459, 66096]],
[145, [42, 6090]],
[146, [411, 60006]],
[147, [47, 6909]],
[148, [45, 6660]],
[149, [651, 96999]],
[150, [4, 600]],
[151, [6, 906]],
[152, [3948, 600096]],
[153, [432, 66096]],
[154, [39, 6006]],
[155, [432, 66960]],
[156, [385, 60060]],
[157, [3828, 600996]],
[158, [57, 9006]],
[159, [44, 6996]],
[160, [6, 960]],
[161, [6, 966]],
[162, [43, 6966]],
[163, [4092, 666996]],
[164, [3714, 609096]],
[165, [4, 660]],
[166, [6, 996]],
[167, [597, 99699]],
[168, [397, 66696]],
[169, [414, 69966]],
[170, [57, 9690]],
[171, [39, 6669]],
[172, [405, 69660]],
[173, [5553, 960669]],
[174, [4, 696]],
[175, [348, 60900]],
[176, [375, 66000]],
[177, [378, 66906]],
[178, [3405, 606090]],
[179, [54, 9666]],
[180, [5, 900]],
[181, [3816, 690696]],
[182, [33, 6006]],
[183, [53, 9699]],
[184, [375, 69000]],
[185, [36, 6660]],
[186, [36, 6696]],
[187, [3738, 699006]],
[188, [3192, 600096]],
[189, [321, 60669]],
[190, [51, 9690]],
[191, [366, 69906]],
[192, [5, 960]],
[193, [342, 66006]],
[194, [4689, 909666]],
[195, [308, 60060]],
[196, [51, 9996]],
[197, [4878, 960966]],
[198, [5, 990]],
[199, [501, 99699]],
[200, [3, 600]],
[201, [299, 60099]],
[202, [3, 606]],
[203, [3, 609]],
[204, [49, 9996]],
[205, [3252, 666660]],
[206, [2916, 600696]],
[207, [338, 69966]],
[208, [462, 96096]],
[209, [474, 99066]],
[210, [29, 6090]],
[211, [28479, 6009069]],
[212, [33, 6996]],
[213, [282, 60066]],
[214, [285, 60990]],
[215, [324, 69660]],
[216, [31, 6696]],
[217, [318, 69006]],
[218, [417, 90906]],
[219, [274, 60006]],
[220, [3, 660]],
[221, [276, 60996]],
[222, [3, 666]],
[223, [3, 669]],
[224, [429, 96096]],
[225, [4, 900]],
[226, [441, 99666]],
[227, [267, 60609]],
[228, [307, 69996]],
[229, [4221, 966609]],
[230, [3, 690]],
[231, [26, 6006]],
[232, [3, 696]],
[233, [3, 699]],
[234, [259, 60606]],
[235, [294, 69090]],
[236, [411, 96996]],
[237, [38, 9006]],
[238, [42, 9996]],
[239, [27894, 6666666]],
[240, [4, 960]],
[241, [249, 60009]],
[242, [273, 66066]],
[243, [272, 66096]],
[244, [2484, 606096]],
[245, [282, 69090]],
[246, [271, 66666]],
[247, [27, 6669]],
[248, [27, 6696]],
[249, [4, 996]],
[250, [24, 6000]],
[251, [3849, 966099]],
[252, [2405, 606060]],
[253, [273, 69069]],
[254, [24, 6096]],
[255, [38, 9690]],
[256, [375, 96000]],
[257, [237, 60909]],
[258, [27, 6966]],
[259, [234, 60606]],
[260, [231, 60060]],
[261, [2336, 609696]],
[262, [2313, 606006]],
[263, [26262, 6906906]],
[264, [25, 6600]],
[265, [264, 69960]],
[266, [2256, 600096]],
[267, [227, 60609]],
[268, [372, 99696]],
[269, [2601, 699669]],
[270, [37, 9990]],
[271, [246, 66666]],
[272, [243, 66096]],
[273, [22, 6006]],
[274, [219, 60006]],
[275, [24, 6600]],
[276, [25, 6900]],
[277, [24078, 6669606]],
[278, [327, 90906]],
[279, [24, 6696]],
[280, [345, 96600]],
[281, [216, 60696]],
[282, [213, 60066]],
[283, [21543, 6096669]],
[284, [2115, 600660]],
[285, [34, 9690]],
[286, [21, 6006]],
[287, [3138, 900606]],
[288, [242, 69696]],
[289, [21, 6069]],
[290, [21, 6090]],
[291, [2289, 666099]],
[292, [2055, 600060]],
[293, [33, 9669]],
[294, [34, 9996]],
[295, [2268, 669060]],
[296, [225, 66600]],
[297, [3367, 999999]],
[298, [2217, 660666]],
[299, [201, 60099]],
[300, [2, 600]],
[301, [2196, 660996]],
[302, [3, 906]],
[303, [2, 606]],
[304, [1974, 600096]],
[305, [318, 96990]],
[306, [216, 66096]],
[307, [228, 69996]],
[308, [195, 60060]],
[309, [311, 96099]],
[310, [216, 66960]],
[311, [309, 96099]],
[312, [308, 96096]],
[313, [192, 60096]],
[314, [1914, 600996]],
[315, [286, 90090]],
[316, [285, 90060]],
[317, [3057, 969069]],
[318, [22, 6996]],
[319, [21, 6699]],
[320, [3, 960]],
[321, [19, 6099]],
[322, [3, 966]],
[323, [3, 969]],
[324, [204, 66096]],
[325, [1848, 600600]],
[326, [2046, 666996]],
[327, [278, 90906]],
[328, [1857, 609096]],
[329, [21, 6909]],
[330, [2, 660]],
[331, [2739, 906609]],
[332, [3, 996]],
[333, [2, 666]],
[334, [2985, 996990]],
[335, [1794, 600990]],
[336, [286, 96096]],
[337, [18, 6066]],
[338, [207, 69966]],
[339, [294, 99666]],
[340, [285, 96900]],
[341, [1956, 666996]],
[342, [193, 66006]],
[343, [17493, 6000099]],
[344, [2025, 696600]],
[345, [2, 690]],
[346, [17361, 6006906]],
[347, [17568, 6096096]],
[348, [2, 696]],
[349, [2751, 960099]],
[350, [174, 60900]],
[351, [19, 6669]],
[352, [198, 69696]],
[353, [1983, 699999]],
[354, [189, 66906]],
[355, [1692, 600660]],
[356, [1941, 690996]],
[357, [17, 6069]],
[358, [27, 9666]],
[359, [1674, 600966]],
[360, [25, 9000]],
[361, [2769, 999609]],
[362, [1908, 690696]],
[363, [182, 66066]],
[364, [165, 60060]],
[365, [1644, 600060]],
[366, [191, 69906]],
[367, [18, 6606]],
[368, [1647, 606096]],
[369, [271, 99999]],
[370, [18, 6660]],
[371, [186, 69006]],
[372, [18, 6696]],
[373, [16263, 6066099]],
[374, [1869, 699006]],
[375, [16, 6000]],
[376, [1596, 600096]],
[377, [1617, 609609]],
[378, [177, 66906]],
[379, [24, 9096]],
[380, [237, 90060]],
[381, [16, 6096]],
[382, [183, 69906]],
[383, [25302, 9690666]],
[384, [25, 9600]],
[385, [156, 60060]],
[386, [171, 66006]],
[387, [18, 6966]],
[388, [15642, 6069096]],
[389, [1791, 696699]],
[390, [154, 60060]],
[391, [156, 60996]],
[392, [255, 99960]],
[393, [1542, 606006]],
[394, [2439, 960966]],
[395, [228, 90060]],
[396, [25, 9900]],
[397, [168, 66696]],
[398, [2505, 996990]],
[399, [1504, 600096]],
[400, [15, 6000]],
[401, [2409, 966009]],
[402, [248, 99696]],
[403, [16602, 6690606]],
[404, [15, 6060]],
[405, [172, 69660]],
[406, [15, 6090]],
[407, [1638, 666666]],
[408, [162, 66096]],
[409, [14841, 6069969]],
[410, [1626, 666660]],
[411, [146, 60006]],
[412, [1458, 600696]],
[413, [162, 66906]],
[414, [169, 69966]],
[415, [24, 9960]],
[416, [231, 96096]],
[417, [218, 90906]],
[418, [237, 99066]],
[419, [16611, 6960009]],
[420, [23, 9660]],
[421, [23019, 9690999]],
[422, [16353, 6900966]],
[423, [142, 60066]],
[424, [165, 69960]],
[425, [228, 96900]],
[426, [141, 60066]],
[427, [1548, 660996]],
[428, [1425, 609900]],
[429, [14, 6006]],
[430, [162, 69660]],
[431, [2229, 960699]],
[432, [153, 66096]],
[433, [15402, 6669066]],
[434, [159, 69006]],
[435, [14, 6090]],
[436, [1536, 669696]],
[437, [13938, 6090906]],
[438, [137, 60006]],
[439, [2064, 906096]],
[440, [15, 6600]],
[441, [226, 99666]],
[442, [138, 60996]],
[443, [20472, 9069096]],
[444, [15, 6660]],
[445, [1362, 606090]],
[446, [15, 6690]],
[447, [217, 96999]],
[448, [2145, 960960]],
[449, [15381, 6906069]],
[450, [2, 900]],
[451, [14649, 6606699]],
[452, [1548, 699696]],
[453, [2, 906]],
[454, [1335, 606090]],
[455, [132, 60060]],
[456, [1316, 600096]],
[457, [19707, 9006099]],
[458, [14412, 6600696]],
[459, [144, 66096]],
[460, [15, 6900]],
[461, [1446, 666606]],
[462, [13, 6006]],
[463, [13173, 6099099]],
[464, [15, 6960]],
[465, [144, 66960]],
[466, [15, 6990]],
[467, [207, 96669]],
[468, [1295, 606060]],
[469, [2061, 966609]],
[470, [147, 69090]],
[471, [1276, 600996]],
[472, [2055, 969960]],
[473, [14163, 6699099]],
[474, [19, 9006]],
[475, [204, 96900]],
[476, [21, 9996]],
[477, [1258, 600066]],
[478, [13947, 6666666]],
[479, [1254, 600666]],
[480, [2, 960]],
[481, [126, 60606]],
[482, [1245, 600090]],
[483, [2, 966]],
[484, [144, 69696]],
[485, [13734, 6660990]],
[486, [136, 66096]],
[487, [12507, 6090909]],
[488, [1242, 606096]],
[489, [1364, 666996]],
[490, [141, 69090]],
[491, [13566, 6660906]],
[492, [1238, 609096]],
[493, [1413, 696609]],
[494, [135, 66690]],
[495, [2, 990]],
[496, [135, 66960]],
[497, [12267, 6096699]],
[498, [2, 996]],
[499, [1401, 699099]],
[500, [12, 6000]],
[501, [199, 99699]],
[502, [18048, 9060096]],
[503, [1332, 669996]],
[504, [1799, 906696]],
[505, [12, 6060]],
[506, [1365, 690690]],
[507, [138, 69966]],
[508, [12, 6096]],
[509, [13674, 6960066]],
[510, [19, 9690]],
[511, [1779, 909069]],
[512, [1308, 669696]],
[513, [13, 6669]],
[514, [1179, 606006]],
[515, [1866, 960990]],
[516, [135, 69660]],
[517, [1857, 960069]],
[518, [117, 60606]],
[519, [1851, 960669]],
[520, [1155, 600600]],
[521, [186, 96906]],
[522, [1168, 609696]],
[523, [1722, 900606]],
[524, [11565, 6060060]],
[525, [116, 60900]],
[526, [13131, 6906906]],
[527, [12537, 6606999]],
[528, [125, 66000]],
[529, [11361, 6009969]],
[530, [132, 69960]],
[531, [126, 66906]],
[532, [1128, 600096]],
[533, [1803, 960999]],
[534, [1135, 606090]],
[535, [114, 60990]],
[536, [186, 99696]],
[537, [18, 9666]],
[538, [1842, 990996]],
[539, [1131, 609609]],
[540, [124, 66960]],
[541, [11109, 6009969]],
[542, [123, 66666]],
[543, [1272, 690696]],
[544, [1215, 660960]],
[545, [1668, 909060]],
[546, [11, 6006]],
[547, [1098, 600606]],
[548, [177, 96996]],
[549, [1094, 600606]],
[550, [12, 6600]],
[551, [12159, 6699609]],
[552, [125, 69000]],
[553, [109602, 60609906]],
[554, [12039, 6669606]],
[555, [12, 6660]],
[556, [1635, 909060]],
[557, [1617, 900669]],
[558, [12, 6696]],
[559, [125061, 69909099]],
[560, [1191, 666960]],
[561, [1246, 699006]],
[562, [108, 60696]],
[563, [12273, 6909699]],
[564, [1064, 600096]],
[565, [1764, 996660]],
[566, [12351, 6990666]],
[567, [107, 60669]],
[568, [10575, 6006600]],
[569, [174, 99006]],
[570, [17, 9690]],
[571, [1746, 996966]],
[572, [105, 60060]],
[573, [122, 69906]],
[574, [1569, 900606]],
[575, [12, 6900]],
[576, [121, 69696]],
[577, [1197, 690669]],
[578, [105, 60690]],
[579, [114, 66006]],
[580, [12, 6960]],
[581, [1716, 996996]],
[582, [1563, 909666]],
[583, [12, 6996]],
[584, [1044, 609696]],
[585, [114, 66690]],
[586, [165, 96690]],
[587, [16518, 9696066]],
[588, [17, 9996]],
[589, [118794, 69969666]],
[590, [1134, 669060]],
[591, [1626, 960966]],
[592, [1125, 666000]],
[593, [162, 96066]],
[594, [11734, 6969996]],
[595, [102, 60690]],
[596, [1626, 969096]],
[597, [167, 99699]],
[598, [102, 60996]],
[599, [11631, 6966969]],
[600, [1, 600]],
[601, [1509, 906909]],
[602, [1098, 660996]],
[603, [1602, 966006]],
[604, [15, 9060]],
[605, [1092, 660660]],
[606, [1, 606]],
[607, [14838, 9006666]],
[608, [987, 600096]],
[609, [1, 609]],
[610, [159, 96990]],
[611, [11406, 6969066]],
[612, [108, 66096]],
[613, [14682, 9000066]],
[614, [114, 69996]],
[615, [1084, 666660]],
[616, [156, 96096]],
[617, [147, 90699]],
[618, [972, 600696]],
[619, [984, 609096]],
[620, [108, 66960]],
[621, [146, 90666]],
[622, [1545, 960990]],
[623, [1122, 699006]],
[624, [154, 96096]],
[625, [96, 60000]],
[626, [96, 60096]],
[627, [158, 99066]],
[628, [957, 600996]],
[629, [954, 600066]],
[630, [143, 90090]],
[631, [15699, 9906069]],
[632, [153, 96696]],
[633, [9493, 6009069]],
[634, [10899, 6909966]],
[635, [96, 60960]],
[636, [11, 6996]],
[637, [957, 609609]],
[638, [105, 66990]],
[639, [94, 60066]],
[640, [15, 9600]],
[641, [156, 99996]],
[642, [95, 60990]],
[643, [9342, 6006906]],
[644, [15, 9660]],
[645, [108, 69660]],
[646, [15, 9690]],
[647, [1068, 690996]],
[648, [102, 66096]],
[649, [1494, 969606]],
[650, [924, 600600]],
[651, [106, 69006]],
[652, [1023, 666996]],
[653, [102, 66606]],
[654, [139, 90906]],
[655, [9252, 6060060]],
[656, [9285, 6090960]],
[657, [137, 90009]],
[658, [105, 69090]],
[659, [10161, 6696099]],
[660, [1, 660]],
[661, [1059, 699999]],
[662, [9213, 6099006]],
[663, [92, 60996]],
[664, [15, 9960]],
[665, [9024, 6000960]],
[666, [1, 666]],
[667, [8997, 6000999]],
[668, [1497, 999996]],
[669, [1, 669]],
[670, [897, 600990]],
[671, [9846, 6606666]],
[672, [143, 96096]],
[673, [9942, 6690966]],
[674, [9, 6066]],
[675, [148, 99900]],
[676, [1035, 699660]],
[677, [8997, 6090969]],
[678, [147, 99666]],
[679, [981, 666099]],
[680, [147, 99960]],
[681, [89, 60609]],
[682, [978, 666996]],
[683, [102, 69666]],
[684, [965, 660060]],
[685, [876, 600060]],
[686, [9621, 6600006]],
[687, [1407, 966609]],
[688, [1017, 699696]],
[689, [13164, 9069996]],
[690, [1, 690]],
[691, [13026, 9000966]],
[692, [14013, 9696996]],
[693, [13, 9009]],
[694, [8784, 6096096]],
[695, [1308, 909060]],
[696, [1, 696]],
[697, [86097, 60009609]],
[698, [13755, 9600990]],
[699, [1, 699]],
[700, [87, 60900]],
[701, [138369, 96996669]],
[702, [95, 66690]],
[703, [1422, 999666]],
[704, [99, 69696]],
[705, [98, 69090]],
[706, [1416, 999696]],
[707, [858, 606606]],
[708, [137, 96996]],
[709, [141, 99969]],
[710, [846, 600660]],
[711, [136, 96696]],
[712, [9705, 6909960]],
[713, [9813, 6996669]],
[714, [14, 9996]],
[715, [84, 60060]],
[716, [135, 96660]],
[717, [9298, 6666666]],
[718, [837, 600966]],
[719, [8484, 6099996]],
[720, [93, 66960]],
[721, [846, 609966]],
[722, [8403, 6066966]],
[723, [83, 60009]],
[724, [954, 690696]],
[725, [84, 60900]],
[726, [91, 66066]],
[727, [8337, 6060999]],
[728, [132, 96096]],
[729, [824, 600696]],
[730, [822, 600060]],
[731, [13626, 9960606]],
[732, [828, 606096]],
[733, [12402, 9090666]],
[734, [9, 6606]],
[735, [94, 69090]],
[736, [1236, 909696]],
[737, [13038, 9609006]],
[738, [1355, 999990]],
[739, [894, 660666]],
[740, [9, 6660]],
[741, [9, 6669]],
[742, [93, 69006]],
[743, [93, 69099]],
[744, [9, 6696]],
[745, [1302, 969990]],
[746, [8976, 6696096]],
[747, [1217, 909099]],
[748, [8952, 6696096]],
[749, [81, 60669]],
[750, [8, 6000]],
[751, [120759, 90690009]],
[752, [798, 600096]],
[753, [1283, 966099]],
[754, [924, 696696]],
[755, [12, 9060]],
[756, [885, 669060]],
[757, [1317, 996969]],
[758, [12, 9096]],
[759, [91, 69069]],
[760, [921, 699960]],
[761, [9159, 6969999]],
[762, [8, 6096]],
[763, [873, 666099]],
[764, [864, 660096]],
[765, [864, 660960]],
[766, [12651, 9690666]],
[767, [8997, 6900699]],
[768, [125, 96000]],
[769, [78114, 60069666]],
[770, [78, 60060]],
[771, [79, 60909]],
[772, [855, 660060]],
[773, [1242, 960066]],
[774, [9, 6966]],
[775, [864, 669600]],
[776, [7821, 6069096]],
[777, [78, 60606]],
[778, [7797, 6066066]],
[779, [771, 600609]],
[780, [77, 60060]],
[781, [84516, 66006996]],
[782, [78, 60996]],
[783, [843, 660069]],
[784, [1275, 999600]],
[785, [7656, 6009960]],
[786, [771, 606006]],
[787, [12207, 9606909]],
[788, [12195, 9609660]],
[789, [8754, 6906906]],
[790, [114, 90060]],
[791, [126, 99666]],
[792, [88, 69696]],
[793, [76533, 60690669]],
[794, [84, 66696]],
[795, [88, 69960]],
[796, [8676, 6906096]],
[797, [867, 690999]],
[798, [752, 600096]],
[799, [1134, 906066]],
[800, [12, 9600]],
[801, [869, 696069]],
[802, [11298, 9060996]],
[803, [822, 660066]],
[804, [124, 99696]],
[805, [12, 9660]],
[806, [8301, 6690606]],
[807, [867, 699669]],
[808, [12, 9696]],
[809, [8241, 6666969]],
[810, [86, 69660]],
[811, [8619, 6990009]],
[812, [75, 60900]],
[813, [82, 66666]],
[814, [819, 666666]],
[815, [8184, 6669960]],
[816, [81, 66096]],
[817, [73557, 60096069]],
[818, [8187, 6696966]],
[819, [11, 9009]],
[820, [813, 666660]],
[821, [7419, 6090999]],
[822, [73, 60006]],
[823, [813, 669099]],
[824, [729, 600696]],
[825, [8, 6600]],
[826, [81, 66906]],
[827, [73278, 60600906]],
[828, [732, 606096]],
[829, [11661, 9666969]],
[830, [12, 9960]],
[831, [8026, 6669606]],
[832, [1155, 960960]],
[833, [12, 9996]],
[834, [109, 90906]],
[835, [1194, 996990]],
[836, [1185, 990660]],
[837, [8, 6696]],
[838, [11457, 9600966]],
[839, [11454, 9609906]],
[840, [115, 96600]],
[841, [7956, 6690996]],
[842, [71373, 60096066]],
[843, [72, 60696]],
[844, [81765, 69009660]],
[845, [828, 699660]],
[846, [71, 60066]],
[847, [78, 66066]],
[848, [825, 699600]],
[849, [7181, 6096669]],
[850, [114, 96900]],
[851, [819, 696969]],
[852, [705, 600660]],
[853, [10632, 9069096]],
[854, [774, 660996]],
[855, [78, 66690]],
[856, [7125, 6099000]],
[857, [1128, 966696]],
[858, [7, 6006]],
[859, [7761, 6666699]],
[860, [81, 69660]],
[861, [1046, 900606]],
[862, [11145, 9606990]],
[863, [6963, 6009069]],
[864, [764, 660096]],
[865, [11106, 9606690]],
[866, [7701, 6669066]],
[867, [7, 6069]],
[868, [795, 690060]],
[869, [114, 99066]],
[870, [7, 6090]],
[871, [69, 60099]],
[872, [768, 669696]],
[873, [763, 666099]],
[874, [6969, 6090906]],
[875, [696, 609000]],
[876, [685, 600060]],
[877, [7878, 6909006]],
[878, [1032, 906096]],
[879, [11, 9669]],
[880, [75, 66000]],
[881, [69126, 60900006]],
[882, [113, 99666]],
[883, [102, 90066]],
[884, [69, 60996]],
[885, [756, 669060]],
[886, [10236, 9069096]],
[887, [74487, 66069969]],
[888, [75, 66600]],
[889, [74991, 66666999]],
[890, [681, 606090]],
[891, [7856, 6999696]],
[892, [75, 66900]],
[893, [672, 600096]],
[894, [739, 660666]],
[895, [108, 96660]],
[896, [7701, 6900096]],
[897, [67, 60099]],
[898, [76905, 69060690]],
[899, [110901, 99699999]],
[900, [1, 900]],
[901, [666, 600066]],
[902, [10653, 9609006]],
[903, [732, 660996]],
[904, [774, 699696]],
[905, [7632, 6906960]],
[906, [1, 906]],
[907, [66258, 60096006]],
[908, [6675, 6060900]],
[909, [1, 909]],
[910, [66, 60060]],
[911, [73179, 66666069]],
[912, [658, 600096]],
[913, [723, 660099]],
[914, [10569, 9660066]],
[915, [106, 96990]],
[916, [7206, 6600696]],
[917, [1047, 960099]],
[918, [72, 66096]],
[919, [7584, 6969696]],
[920, [75, 69000]],
[921, [76, 69996]],
[922, [723, 666606]],
[923, [1083, 999609]],
[924, [65, 60060]],
[925, [72, 66600]],
[926, [65541, 60690966]],
[927, [648, 600696]],
[928, [75, 69600]],
[929, [10731, 9969099]],
[930, [72, 66960]],
[931, [70899, 66006969]],
[932, [75, 69900]],
[933, [103, 96099]],
[934, [1035, 966690]],
[935, [7476, 6990060]],
[936, [1036, 969696]],
[937, [105657, 99000609]],
[938, [10242, 9606996]],
[939, [64, 60096]],
[940, [639, 600660]],
[941, [966, 909006]],
[942, [638, 600996]],
[943, [10563, 9960909]],
[944, [1059, 999696]],
[945, [642, 606690]],
[946, [10221, 9669066]],
[947, [10527, 9969069]],
[948, [95, 90060]],
[949, [7281, 6909669]],
[950, [102, 96900]],
[951, [1019, 969069]],
[952, [105, 99960]],
[953, [702, 669006]],
[954, [629, 600066]],
[955, [732, 699060]],
[956, [69666, 66600696]],
[957, [7, 6699]],
[958, [627, 600666]],
[959, [6351, 6090609]],
[960, [1, 960]],
[961, [6306, 6060066]],
[962, [63, 60606]],
[963, [63, 60669]],
[964, [6225, 6000900]],
[965, [684, 660060]],
[966, [1, 966]],
[967, [6888, 6660696]],
[968, [72, 69696]],
[969, [1, 969]],
[970, [6867, 6660990]],
[971, [69, 66999]],
[972, [68, 66096]],
[973, [6852, 6666996]],
[974, [9309, 9066966]],
[975, [616, 600600]],
[976, [621, 606096]],
[977, [68178, 66609906]],
[978, [682, 666996]],
[979, [681, 666699]],
[980, [102, 99960]],
[981, [679, 666099]],
[982, [6783, 6660906]],
[983, [702, 690066]],
[984, [619, 609096]],
[985, [9756, 9609660]],
[986, [6786, 6690996]],
[987, [7, 6909]],
[988, [675, 666900]],
[989, [61281, 60606909]],
[990, [1, 990]],
[991, [666, 660006]],
[992, [675, 669600]],
[993, [913, 906609]],
[994, [61335, 60966990]],
[995, [1002, 996990]],
[996, [1, 996]],
[997, [602778, 600969666]],
[998, [1002, 999996]],
[999, [1, 999]],
[1000, [6, 6000]]]
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]],
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A few more notes: It's clear looking for products with digits [n,0] will be (more or less) just n-fold larger than the results for [1,0]. It's also clear there's no way to use pure multiplication for all pairs of digits; for example, [2,3] is a no-go—plenty of numbers you can't force into a product with only 2s and 3s in it.
But it's not clear to me (yet) whether any other pairs of the possibilities might also work, nor which triplets will work, and so on. For example, [012] should produce smaller multipliers... right? Maybe?