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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 |
# 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 |
One interesting thing this list provokes, beyond the obvious genetic programming project: I see three places where adjacent pairs of numbers in the left column produce adjacent pairs in the right column.
- "7 * 143 = 1001" and "8 * 125 = 1000"
- "70 * 143 = 10010" & "71 * 141 = 10011"
- "96 * 115625 = 11100000" & "97 * 114433 = 11100001"
and that's eye-catching, and not immediately intuitive. But I ain't any kind of Number Theory person.
Another question, related, is whether there are adjacent pairs of the same sort where the Hamming distance between the products is only 1, without necessarily being the last place. I don't see any upon brief inspection; I see a free 2- and 3-digit differences though.
results of an overnight run
"1: 1 * 1 = 1"
"2: 10 * 1 = 10"
"3: 11 * 1 = 11"
"4: 100 * 1 = 100"
"5: 101 * 1 = 101"
"6: 110 * 1 = 110"
"7: 111 * 1 = 111"
"8: 1000 * 1 = 1000"
"9: 5 * 2 = 10"
"10: 50 * 2 = 100"
"11: 55 * 2 = 110"
"12: 500 * 2 = 1000"
"13: 505 * 2 = 1010"
"14: 550 * 2 = 1100"
"15: 555 * 2 = 1110"
"16: 37 * 3 = 111"
"17: 337 * 3 = 1011"
"18: 367 * 3 = 1101"
"19: 370 * 3 = 1110"
"20: 25 * 4 = 100"
"21: 250 * 4 = 1000"
"22: 275 * 4 = 1100"
"23: 2 * 5 = 10"
"24: 20 * 5 = 100"
"25: 22 * 5 = 110"
"26: 200 * 5 = 1000"
"27: 202 * 5 = 1010"
"28: 220 * 5 = 1100"
"29: 222 * 5 = 1110"
"30: 185 * 6 = 1110"
"31: 143 * 7 = 1001"
"32: 125 * 8 = 1000"
"33: 91 * 11 = 1001"
"34: 910 * 11 = 10010"
"35: 925 * 12 = 11100"
"36: 77 * 13 = 1001"
"37: 770 * 13 = 10010"
"38: 777 * 13 = 10101"
"39: 847 * 13 = 11011"
"40: 715 * 14 = 10010"
"41: 74 * 15 = 1110"
"42: 674 * 15 = 10110"
"43: 734 * 15 = 11010"
"44: 740 * 15 = 11100"
"45: 625 * 16 = 10000"
"46: 653 * 17 = 11101"
"47: 579 * 19 = 11001"
"48: 481 * 21 = 10101"
"49: 455 * 22 = 10010"
"50: 4 * 25 = 100"
"51: 40 * 25 = 1000"
"52: 44 * 25 = 1100"
"53: 400 * 25 = 10000"
"54: 404 * 25 = 10100"
"55: 440 * 25 = 11000"
"56: 444 * 25 = 11100"
"57: 385 * 26 = 10010"
"58: 286 * 35 = 10010"
"59: 3 * 37 = 111"
"60: 30 * 37 = 1110"
"61: 273 * 37 = 10101"
"62: 300 * 37 = 11100"
"63: 259 * 39 = 10101"
"64: 271 * 41 = 11111"
"65: 213 * 47 = 10011"
"66: 182 * 55 = 10010"
"67: 193 * 57 = 11001"
"68: 154 * 65 = 10010"
"69: 141 * 71 = 10011"
"70: 137 * 73 = 10001"
"71: 15 * 74 = 1110"
"72: 150 * 74 = 11100"
"73: 148 * 75 = 11100"
"74: 13 * 77 = 1001"
"75: 130 * 77 = 10010"
"76: 121 * 91 = 11011"
"77: 962 * 105 = 101010"
"78: 901 * 111 = 100011"
"79: 991 * 111 = 110001"
"80: 965 * 114 = 110010"
"81: 8 * 125 = 1000"
"82: 80 * 125 = 10000"
"83: 88 * 125 = 11000"
"84: 800 * 125 = 100000"
"85: 808 * 125 = 101000"
"86: 880 * 125 = 110000"
"87: 888 * 125 = 111000"
"88: 771 * 131 = 101001"
"89: 73 * 137 = 10001"
"90: 730 * 137 = 100010"
"91: 803 * 137 = 110011"
"92: 71 * 141 = 10011"
"93: 710 * 141 = 100110"
"94: 705 * 142 = 100110"
"95: 7 * 143 = 1001"
"96: 70 * 143 = 10010"
"97: 700 * 143 = 100100"
"98: 707 * 143 = 101101"
"99: 685 * 146 = 100010"
"100: 75 * 148 = 11100"
"101: 750 * 148 = 111000"
"102: 739 * 149 = 110111"
"103: 65 * 154 = 10010"
"104: 650 * 154 = 100100"
"105: 629 * 159 = 100011"
"106: 572 * 175 = 100100"
"107: 605 * 182 = 110110"
"108: 547 * 183 = 100101"
"109: 6 * 185 = 1110"
"110: 60 * 185 = 11100"
"111: 546 * 185 = 101010"
"112: 600 * 185 = 111000"
"113: 57 * 193 = 11001"
"114: 570 * 193 = 110010"
"115: 518 * 195 = 101010"
"116: 542 * 205 = 111110"
"117: 479 * 209 = 100111"
"118: 47 * 213 = 10011"
"119: 470 * 213 = 100110"
"120: 426 * 235 = 100110"
"121: 461 * 241 = 111101"
"122: 393 * 257 = 101001"
"123: 39 * 259 = 10101"
"124: 390 * 259 = 101010"
"125: 429 * 259 = 111111"
"126: 41 * 271 = 11111"
"127: 410 * 271 = 111110"
"128: 407 * 273 = 111111"
"129: 365 * 274 = 100010"
"130: 364 * 275 = 100100"
"131: 355 * 282 = 100110"
"132: 386 * 285 = 110010"
"133: 35 * 286 = 10010"
"134: 350 * 286 = 100100"
"135: 375 * 296 = 111000"
"136: 325 * 308 = 100100"
"137: 308 * 325 = 100100"
"138: 282 * 355 = 100110"
"139: 274 * 365 = 100010"
"140: 296 * 375 = 111000"
"141: 26 * 385 = 10010"
"142: 260 * 385 = 100100"
"143: 285 * 386 = 110010"
"144: 257 * 393 = 101001"
"145: 235 * 426 = 100110"
"146: 242 * 455 = 110110"
"147: 241 * 461 = 111101"
"148: 209 * 479 = 100111"
"149: 21 * 481 = 10101"
"150: 210 * 481 = 101010"
"151: 231 * 481 = 111111"
"152: 195 * 518 = 101010"
"153: 205 * 542 = 111110"
"154: 183 * 547 = 100101"
"155: 175 * 572 = 100100"
"156: 19 * 579 = 11001"
"157: 190 * 579 = 110010"
"158: 16 * 625 = 10000"
"159: 160 * 625 = 100000"
"160: 176 * 625 = 110000"
"161: 159 * 629 = 100011"
"162: 17 * 653 = 11101"
"163: 170 * 653 = 111010"
"164: 146 * 685 = 100010"
"165: 142 * 705 = 100110"
"166: 14 * 715 = 10010"
"167: 140 * 715 = 100100"
"168: 149 * 739 = 110111"
"169: 131 * 771 = 101001"
"170: 12 * 925 = 11100"
"171: 120 * 925 = 111000"
"172: 105 * 962 = 101010"
"173: 114 * 965 = 110010"
"174: 971 * 1031 = 1001101"
"175: 975 * 1036 = 1010100"
"176: 958 * 1045 = 1001110"
"177: 961 * 1051 = 1010011"
"178: 94 * 1065 = 100110"
"179: 940 * 1065 = 1001100"
"180: 915 * 1094 = 1001010"
"181: 875 * 1144 = 1001000"
"182: 95 * 1158 = 110010"
"183: 950 * 1158 = 1100100"
"184: 852 * 1175 = 1001100"
"185: 922 * 1205 = 1111010"
"186: 83 * 1217 = 101011"
"187: 830 * 1217 = 1010110"
"188: 883 * 1247 = 1101101"
"189: 795 * 1258 = 1000110"
"190: 786 * 1285 = 1010010"
"191: 78 * 1295 = 101010"
"192: 780 * 1295 = 1010100"
"193: 858 * 1295 = 1111110"
"194: 85 * 1306 = 111010"
"195: 850 * 1306 = 1110100"
"196: 82 * 1355 = 111110"
"197: 820 * 1355 = 1111100"
"198: 814 * 1365 = 1111110"
"199: 728 * 1375 = 1001000"
"200: 772 * 1425 = 1100100"
"201: 745 * 1478 = 1101110"
"202: 655 * 1542 = 1010010"
"203: 616 * 1625 = 1001000"
"204: 61 * 1641 = 100101"
"205: 610 * 1641 = 1001010"
"206: 671 * 1641 = 1101111"
"207: 583 * 1717 = 1001011"
"208: 564 * 1775 = 1001100"
"209: 617 * 1783 = 1100111"
"210: 548 * 1825 = 1000100"
"211: 592 * 1875 = 1110000"
"212: 53 * 1887 = 100011"
"213: 530 * 1887 = 1000110"
"214: 533 * 1897 = 1011101"
"215: 525 * 1924 = 1010100"
"216: 52 * 1925 = 100100"
"217: 520 * 1925 = 1001000"
"218: 51 * 1961 = 100011"
"219: 510 * 1961 = 1000110"
"220: 514 * 1965 = 1010010"
"221: 557 * 1993 = 1110101"
"222: 484 * 2275 = 1101100"
"223: 482 * 2305 = 1111010"
"224: 475 * 2316 = 1100100"
"225: 418 * 2395 = 1001110"
"226: 42 * 2405 = 101010"
"227: 420 * 2405 = 1010100"
"228: 462 * 2405 = 1111110"
"229: 415 * 2434 = 1010110"
"230: 389 * 2599 = 1011011"
"231: 425 * 2612 = 1110100"
"232: 411 * 2701 = 1110111"
"233: 366 * 2735 = 1001010"
"234: 38 * 2895 = 110010"
"235: 380 * 2895 = 1100100"
"236: 32 * 3125 = 100000"
"237: 320 * 3125 = 1000000"
"238: 352 * 3125 = 1100000"
"239: 318 * 3145 = 1000110"
"240: 343 * 3207 = 1100001"
"241: 34 * 3265 = 111010"
"242: 340 * 3265 = 1110100"
"243: 305 * 3282 = 1001010"
"244: 303 * 3337 = 1011111"
"245: 33 * 3367 = 111111"
"246: 330 * 3367 = 1111110"
"247: 292 * 3425 = 1000100"
"248: 287 * 3523 = 1011101"
"249: 284 * 3525 = 1001100"
"250: 28 * 3575 = 100100"
"251: 280 * 3575 = 1001000"
"252: 31 * 3581 = 111011"
"253: 310 * 3581 = 1110110"
"254: 298 * 3695 = 1101110"
"255: 265 * 3774 = 1000110"
"256: 262 * 3855 = 1010010"
"257: 255 * 3922 = 1000110"
"258: 24 * 4625 = 111000"
"259: 240 * 4625 = 1110000"
"260: 239 * 4649 = 1111111"
"261: 23 * 4787 = 110101"
"262: 230 * 4787 = 1101010"
"263: 228 * 4825 = 1100100"
"264: 219 * 5069 = 1110111"
"265: 188 * 5325 = 1001100"
"266: 187 * 5353 = 1001011"
"267: 181 * 5531 = 1001111"
"268: 166 * 6085 = 1010110"
"269: 163 * 6197 = 1010111"
"270: 156 * 6475 = 1010100"
"271: 165 * 6734 = 1111110"
"272: 164 * 6775 = 1111100"
"273: 155 * 7162 = 1110110"
"274: 151 * 7351 = 1110001"
"275: 147 * 7483 = 1100001"
"276: 122 * 8205 = 1001010"
"277: 113 * 8947 = 1011011"
"278: 106 * 9435 = 1000110"
"279: 115 * 9574 = 1101010"
"280: 104 * 9625 = 1001000"
"281: 102 * 9805 = 1000110"
"282: 959 * 10439 = 10011001"
"283: 949 * 10549 = 10011001"
"284: 935 * 10706 = 10010110"
"285: 905 * 11062 = 10011110"
"286: 968 * 11375 = 11011000"
"287: 869 * 11519 = 10010011"
"288: 964 * 11525 = 11110100"
"289: 851 * 11751 = 10000101"
"290: 836 * 11975 = 10011100"
"291: 84 * 12025 = 1010100"
"292: 840 * 12025 = 10101000"
"293: 924 * 12025 = 11111100"
"294: 815 * 12394 = 10101110"
"295: 778 * 12995 = 10110110"
"296: 829 * 13269 = 11000001"
"297: 825 * 13468 = 11111100"
"298: 822 * 13505 = 11101110"
"299: 743 * 13607 = 10110001"
"300: 732 * 13675 = 10010100"
"301: 775 * 14324 = 11101100"
"302: 697 * 14363 = 10011011"
"303: 703 * 14367 = 10100001"
"304: 76 * 14475 = 1100100"
"305: 760 * 14475 = 11001000"
"306: 759 * 14639 = 11111001"
"307: 755 * 14702 = 11100010"
"308: 735 * 14966 = 11000010"
"309: 651 * 15361 = 10000011"
"310: 64 * 15625 = 1000000"
"311: 640 * 15625 = 10000000"
"312: 704 * 15625 = 11000000"
"313: 636 * 15725 = 10001100"
"314: 637 * 15873 = 10111101"
"315: 686 * 16035 = 11000010"
"316: 68 * 16325 = 1110100"
"317: 680 * 16325 = 11101000"
"318: 67 * 16433 = 1101011"
"319: 670 * 16433 = 11010110"
"320: 606 * 16685 = 10111110"
"321: 66 * 16835 = 1111110"
"322: 660 * 16835 = 11111100"
"323: 587 * 17053 = 10010111"
"324: 584 * 17125 = 10001000"
"325: 574 * 17615 = 10111010"
"326: 568 * 17625 = 10011000"
"327: 56 * 17875 = 1001000"
"328: 560 * 17875 = 10010000"
"329: 565 * 17894 = 10110110"
"330: 62 * 17905 = 1110110"
"331: 620 * 17905 = 11101100"
"332: 609 * 18079 = 11010111"
"333: 596 * 18475 = 11011100"
"334: 541 * 18671 = 10101011"
"335: 539 * 18759 = 10111101"
"336: 575 * 19148 = 11010100"
"337: 524 * 19275 = 10100100"
"338: 511 * 19591 = 10011001"
"339: 507 * 19943 = 10111101"
"340: 537 * 20503 = 11010111"
"341: 497 * 22133 = 11000101"
"342: 49 * 22449 = 1100001"
"343: 490 * 22449 = 11000010"
"344: 433 * 23097 = 10001001"
"345: 48 * 23125 = 1110000"
"346: 480 * 23125 = 11100000"
"347: 478 * 23245 = 11111110"
"348: 419 * 23869 = 10001111"
"349: 46 * 23935 = 1101010"
"350: 460 * 23935 = 11010100"
"351: 456 * 24125 = 11001000"
"352: 438 * 25345 = 11101110"
"353: 437 * 25403 = 11101111"
"354: 43 * 25607 = 1101101"
"355: 430 * 25607 = 11011010"
"356: 376 * 26625 = 10011000"
"357: 374 * 26765 = 10010110"
"358: 362 * 27655 = 10011110"
"359: 397 * 27733 = 11010001"
"360: 383 * 28747 = 11010101"
"361: 377 * 29443 = 11100011"
"362: 332 * 30425 = 10101100"
"363: 361 * 30751 = 11101111"
"364: 326 * 30985 = 10101110"
"365: 341 * 32261 = 11001001"
"366: 312 * 32375 = 10101000"
"367: 335 * 32866 = 11010110"
"368: 329 * 33769 = 11110001"
"369: 328 * 33875 = 11111000"
"370: 323 * 34087 = 11010101"
"371: 313 * 35147 = 11001011"
"372: 302 * 36755 = 11100010"
"373: 294 * 37415 = 11000010"
"374: 29 * 37969 = 1101101"
"375: 290 * 37969 = 11011010"
"376: 263 * 38027 = 10001101"
"377: 244 * 41025 = 10010100"
"378: 253 * 43917 = 11111001"
"379: 226 * 44735 = 10110110"
"380: 245 * 44898 = 11000010"
"381: 217 * 46083 = 10000011"
"382: 212 * 47175 = 10001100"
"383: 208 * 48125 = 10010000"
"384: 227 * 48463 = 11001101"
"385: 204 * 49025 = 10001100"
"386: 201 * 49801 = 10010001"
"387: 215 * 51214 = 11011010"
"388: 203 * 54237 = 11010111"
"389: 191 * 58121 = 11101111"
"390: 169 * 59829 = 10111101"
"391: 168 * 60125 = 10101000"
"392: 179 * 61509 = 11010111"
"393: 167 * 66533 = 11111011"
"394: 161 * 68951 = 11101111"
"395: 152 * 72375 = 11001000"
"396: 145 * 75938 = 11011010"
"397: 128 * 78125 = 10000000"
"398: 136 * 81625 = 11101000"
"399: 134 * 82165 = 11010110"
"400: 133 * 83467 = 11101111"
"401: 132 * 84175 = 11111100"
"402: 127 * 86693 = 11010011"
"403: 112 * 89375 = 10010000"
"404: 124 * 89525 = 11101100"
"405: 941 * 106271 = 100001011"
"406: 93 * 107527 = 10000011"
"407: 930 * 107527 = 100000110"
"408: 103 * 107767 = 11100001"
"409: 994 * 110665 = 110001010"
"410: 98 * 112245 = 11000010"
"411: 980 * 112245 = 110000100"
"412: 881 * 113621 = 100100101"
"413: 97 * 114433 = 11100001"
"414: 970 * 114433 = 111000010"
"415: 866 * 115485 = 100010010"
"416: 96 * 115625 = 11100000"
"417: 960 * 115625 = 111000000"
"418: 957 * 116093 = 111101001"
"419: 956 * 116225 = 111111100"
"420: 955 * 116242 = 111011110"
"421: 838 * 119345 = 100011110"
"422: 845 * 119658 = 101111010"
"423: 92 * 119675 = 11010100"
"424: 920 * 119675 = 110101000"
"425: 912 * 120625 = 110010000"
"426: 895 * 123018 = 110101110"
"427: 89 * 123709 = 11010101"
"428: 890 * 123709 = 110101010"
"429: 87 * 126553 = 11010111"
"430: 870 * 126553 = 110101110"
"431: 79 * 126709 = 10010011"
"432: 790 * 126709 = 100100110"
"433: 876 * 126725 = 111011100"
"434: 874 * 127015 = 111011110"
"435: 871 * 127441 = 111001111"
"436: 86 * 128035 = 11011010"
"437: 860 * 128035 = 110110100"
"438: 835 * 133066 = 111110110"
"439: 752 * 133125 = 100110000"
"440: 748 * 133825 = 100101100"
"441: 805 * 137902 = 111011110"
"442: 724 * 138275 = 100111100"
"443: 794 * 138665 = 110100010"
"444: 766 * 143735 = 110101010"
"445: 69 * 144929 = 10000101"
"446: 690 * 144929 = 100001010"
"447: 689 * 145299 = 100111011"
"448: 754 * 147215 = 111000110"
"449: 673 * 148737 = 100100001"
"450: 737 * 149403 = 110110011"
"451: 663 * 150997 = 100111011"
"452: 725 * 151876 = 110110100"
"453: 664 * 152125 = 101011000"
"454: 722 * 153755 = 111011110"
"455: 649 * 154239 = 100101111"
"456: 652 * 154925 = 101011100"
"457: 631 * 160081 = 101011111"
"458: 682 * 161305 = 110010010"
"459: 624 * 161875 = 101010000"
"460: 665 * 166934 = 111011110"
"461: 658 * 168845 = 111100010"
"462: 656 * 169375 = 111110000"
"463: 646 * 170435 = 110101010"
"464: 635 * 173386 = 110100110"
"465: 626 * 175735 = 110010110"
"466: 627 * 177193 = 111100011"
"467: 561 * 178451 = 100111011"
"468: 613 * 179447 = 110001011"
"469: 604 * 183775 = 111000100"
"470: 59 * 186629 = 11011111"
"471: 590 * 186629 = 110111110"
"472: 588 * 187075 = 110000100"
"473: 58 * 189845 = 11011010"
"474: 580 * 189845 = 110110100"
"475: 526 * 190135 = 100011010"
"476: 577 * 190643 = 110001011"
"477: 553 * 198917 = 110001101"
"478: 501 * 199601 = 100000101"
"479: 489 * 204499 = 100000011"
"480: 488 * 205125 = 100101000"
"481: 483 * 209317 = 101100111"
"482: 467 * 214133 = 100000111"
"483: 465 * 215054 = 100000110"
"484: 515 * 215534 = 111000010"
"485: 506 * 219585 = 111110010"
"486: 452 * 223675 = 101101100"
"487: 493 * 225357 = 111101001"
"488: 485 * 228866 = 111000010"
"489: 434 * 230415 = 100000110"
"490: 431 * 234571 = 101100101"
"491: 473 * 234907 = 111111011"
"492: 424 * 235875 = 100011000"
"493: 421 * 237791 = 100110011"
"494: 463 * 237797 = 110100011"
"495: 416 * 240625 = 100100000"
"496: 454 * 242315 = 110011010"
"497: 408 * 245125 = 100011000"
"498: 445 * 247418 = 110101010"
"499: 449 * 247439 = 111100111"
"500: 443 * 248307 = 110000001"
"501: 402 * 249005 = 100100010"
"502: 435 * 253106 = 110101110"
"503: 395 * 253418 = 100100110"
"504: 417 * 264053 = 110110101"
"505: 379 * 264119 = 100101101"
"506: 381 * 265121 = 101011101"
"507: 371 * 269841 = 100111011"
"508: 373 * 270807 = 101011011"
"509: 406 * 271185 = 110101110"
"510: 403 * 275437 = 111001111"
"511: 357 * 280423 = 100111011"
"512: 345 * 289858 = 100001010"
"513: 382 * 290605 = 111011110"
"514: 338 * 299145 = 101111010"
"515: 336 * 300625 = 101010000"
"516: 358 * 307545 = 110101110"
"517: 311 * 324791 = 101010001"
"518: 334 * 332665 = 111110110"
"519: 333 * 333667 = 111111111"
"520: 331 * 335381 = 111011111"
"521: 321 * 342991 = 110100111"
"522: 291 * 344021 = 100110111"
"523: 322 * 344755 = 111011110"
"524: 319 * 348279 = 111101001"
"525: 281 * 359431 = 101000111"
"526: 304 * 361875 = 110010000"
"527: 269 * 372119 = 100100011"
"528: 295 * 373258 = 110111110"
"529: 293 * 375427 = 110000111"
"530: 256 * 390625 = 100000000"
"531: 277 * 397513 = 110111101"
"532: 272 * 408125 = 111010000"
"533: 268 * 410825 = 110101100"
"534: 266 * 417335 = 111011110"
"535: 264 * 420875 = 111111000"
"536: 237 * 426203 = 101010111"
"537: 254 * 433465 = 110100110"
"538: 251 * 438251 = 110001001"
"539: 224 * 446875 = 100100000"
"540: 248 * 447625 = 111011000"
"541: 221 * 452991 = 100111011"
"542: 233 * 476867 = 111110011"
"543: 211 * 521801 = 110100011"
"544: 186 * 537635 = 100000110"
"545: 206 * 538835 = 111000010"
"546: 199 * 557789 = 111000011"
"547: 196 * 561225 = 110000100"
"548: 177 * 565543 = 100101111"
"549: 194 * 572165 = 111000010"
"550: 192 * 578125 = 111000000"
"551: 184 * 598375 = 110101000"
"552: 178 * 618545 = 110101010"
"553: 174 * 632765 = 110101110"
"554: 158 * 633545 = 100100110"
"555: 172 * 640175 = 110110100"
"556: 138 * 724645 = 100001010"
"557: 129 * 775969 = 100100001"
"558: 139 * 792159 = 110110101"
"559: 119 * 841269 = 100111011"
"560: 118 * 933145 = 110111110"
"561: 107 * 934673 = 100010011"
"562: 116 * 949225 = 110110100"
"563: 993 * 1018127 = 1011000111"
"564: 978 * 1022495 = 1000000110"
"565: 976 * 1025625 = 1001010000"
"566: 977 * 1034913 = 1011110001"
"567: 973 * 1038037 = 1010010001"
"568: 966 * 1046585 = 1011001110"
"569: 943 * 1061507 = 1001001101"
"570: 947 * 1067583 = 1011001101"
"571: 934 * 1070665 = 1000001110"
"572: 995 * 1115578 = 1110000110"
"573: 904 * 1118375 = 1011011000"
"574: 979 * 1123709 = 1100111111"
"575: 986 * 1126785 = 1111010010"
"576: 885 * 1131086 = 1001011110"
"577: 981 * 1132631 = 1111111011"
"578: 879 * 1138909 = 1001101011"
"579: 868 * 1152075 = 1000001100"
"580: 939 * 1171459 = 1100000001"
"581: 862 * 1172855 = 1011001010"
"582: 853 * 1173517 = 1001010001"
"583: 946 * 1174535 = 1111110110"
"584: 848 * 1179375 = 1000110000"
"585: 929 * 1184069 = 1100000101"
"586: 842 * 1188955 = 1001100110"
"587: 926 * 1188985 = 1101000110"
"588: 843 * 1199407 = 1011100101"
"589: 832 * 1203125 = 1001000000"
"590: 908 * 1211575 = 1100110100"
"591: 833 * 1212497 = 1010010001"
"592: 913 * 1216977 = 1111100001"
"593: 816 * 1225625 = 1000110000"
"594: 903 * 1229237 = 1110001011"
"595: 811 * 1234291 = 1001010001"
"596: 898 * 1237195 = 1111001110"
"597: 886 * 1241535 = 1100000010"
"598: 804 * 1245025 = 1001000100"
"599: 807 * 1251673 = 1010100111"
"600: 801 * 1262311 = 1011111111"
"601: 793 * 1273657 = 1010010001"
"602: 781 * 1281831 = 1001110011"
"603: 779 * 1297819 = 1011001001"
"604: 834 * 1320265 = 1101101010"
"605: 758 * 1320595 = 1001011010"
"606: 762 * 1325605 = 1010111010"
"607: 823 * 1336587 = 1100011101"
"608: 742 * 1349205 = 1001110110"
"609: 746 * 1354035 = 1010110110"
"610: 812 * 1355925 = 1101011100"
"611: 806 * 1377185 = 1110011110"
"612: 799 * 1378099 = 1101101101"
"613: 714 * 1402115 = 1001110110"
"614: 699 * 1432189 = 1001100111"
"615: 764 * 1453025 = 1110111100"
"616: 763 * 1454797 = 1110010111"
"617: 753 * 1474117 = 1110010101"
"618: 751 * 1479361 = 1111000111"
"619: 676 * 1495725 = 1011110100"
"620: 669 * 1496429 = 1001111001"
"621: 672 * 1503125 = 1010100000"
"622: 731 * 1506171 = 1101011001"
"623: 716 * 1537725 = 1101011100"
"624: 721 * 1539681 = 1110110001"
"625: 713 * 1542777 = 1100000001"
"626: 645 * 1551938 = 1001000010"
"627: 695 * 1584318 = 1101101010"
"628: 622 * 1623955 = 1010100010"
"629: 679 * 1634919 = 1110110001"
"630: 668 * 1663325 = 1111101100"
"631: 667 * 1664333 = 1110110111"
"632: 666 * 1668335 = 1111111110"
"633: 662 * 1676905 = 1110111110"
"634: 595 * 1682538 = 1001110110"
"635: 593 * 1688027 = 1001000011"
"636: 642 * 1714955 = 1101001110"
"637: 582 * 1720105 = 1001101110"
"638: 644 * 1723775 = 1110111100"
"639: 641 * 1731671 = 1110001111"
"640: 638 * 1741395 = 1111010010"
"641: 571 * 1751331 = 1000010001"
"642: 623 * 1767257 = 1101001111"
"643: 569 * 1776819 = 1011010011"
"644: 562 * 1797155 = 1010001110"
"645: 611 * 1800491 = 1100100001"
"646: 608 * 1809375 = 1100100000"
"647: 551 * 1833051 = 1010011101"
"648: 597 * 1842563 = 1100010111"
"649: 603 * 1842637 = 1111110111"
"650: 538 * 1860595 = 1001000110"
"651: 535 * 1869346 = 1000100110"
"652: 586 * 1877135 = 1100001110"
"653: 529 * 1890569 = 1000111001"
"654: 517 * 1936383 = 1001110011"
"655: 521 * 1938791 = 1010110111"
"656: 519 * 1947979 = 1011001101"
"657: 512 * 1953125 = 1000000000"
"658: 563 * 1955597 = 1101001111"
"659: 554 * 1987565 = 1101111010"
"660: 544 * 2040625 = 1110100000"
"661: 536 * 2054125 = 1101011000"
"662: 532 * 2086675 = 1110111100"
"663: 527 * 2089393 = 1101110111"
"664: 528 * 2104375 = 1111110000"
"665: 474 * 2131015 = 1010101110"
"666: 508 * 2167325 = 1101001100"
"667: 503 * 2188867 = 1101000101"
"668: 457 * 2190593 = 1001101001"
"669: 502 * 2191255 = 1100010010"
"670: 448 * 2234375 = 1001000000"
"671: 496 * 2238125 = 1110110000"
"672: 487 * 2258953 = 1100110111"
"673: 442 * 2264955 = 1001110110"
"674: 469 * 2345629 = 1100100001"
"675: 427 * 2365363 = 1010010001"
"676: 466 * 2384335 = 1111100110"
"677: 451 * 2463661 = 1111111111"
"678: 422 * 2609005 = 1101000110"
"679: 372 * 2688175 = 1000001100"
"680: 409 * 2689489 = 1100001001"
"681: 412 * 2694175 = 1110000100"
"682: 363 * 2755127 = 1000111101"
"683: 401 * 2770601 = 1111011001"
"684: 399 * 2784489 = 1111011111"
"685: 359 * 2788329 = 1001010111"
"686: 398 * 2788945 = 1110000110"
"687: 392 * 2806125 = 1100001000"
"688: 354 * 2827715 = 1001011110"
"689: 388 * 2860825 = 1110000100"
"690: 384 * 2890625 = 1110000000"
"691: 347 * 2910983 = 1010111101"
"692: 339 * 2979649 = 1010101011"
"693: 368 * 2991875 = 1101010000"
"694: 327 * 3061193 = 1001010111"
"695: 356 * 3092725 = 1101010100"
"696: 349 * 3152149 = 1100100001"
"697: 348 * 3163825 = 1101011100"
"698: 316 * 3167725 = 1001001100"
"699: 344 * 3200875 = 1101101000"
"700: 309 * 3592589 = 1110110001"
"701: 276 * 3623225 = 1000010100"
"702: 301 * 3657811 = 1101001111"
"703: 267 * 3786933 = 1011111111"
"704: 289 * 3806609 = 1100110001"
"705: 258 * 3879845 = 1001000010"
"706: 278 * 3960795 = 1101101010"
"707: 247 * 4048583 = 1000000001"
"708: 238 * 4206345 = 1001110110"
"709: 249 * 4462249 = 1111100001"
"710: 223 * 4489287 = 1001111001"
"711: 236 * 4665725 = 1101111100"
"712: 214 * 4673365 = 1000100110"
"713: 232 * 4746125 = 1101101000"
"714: 197 * 5588833 = 1101000101"
"715: 173 * 5843937 = 1011001101"
"716: 171 * 6497141 = 1111011111"
"717: 109 * 9183579 = 1001010111"
"718: 967 * 10351603 = 10010000101"
"719: 948 * 10655075 = 10101011100"
"720: 937 * 10789873 = 10110111001"
"721: 923 * 10942687 = 10100100101"
"722: 914 * 10952965 = 10011010010"
"723: 917 * 11025083 = 10110001111"
"724: 907 * 11026473 = 10001011011"
"725: 897 * 11159533 = 10010101101"
"726: 896 * 11171875 = 10010000000"
"727: 985 * 11177666 = 11010001010"
"728: 983 * 11190347 = 11000111101"
"729: 992 * 11190625 = 11101100000"
"730: 974 * 11294765 = 11001101110"
"731: 884 * 11324775 = 10011101100"
"732: 857 * 11680293 = 10010011101"
"733: 865 * 11687874 = 10110011010"
"734: 938 * 11728145 = 11001000010"
"735: 849 * 11791649 = 10011110001"
"736: 854 * 11826815 = 10100100010"
"737: 933 * 11898297 = 11101111101"
"738: 932 * 11921675 = 11111001100"
"739: 821 * 12180391 = 10000101011"
"740: 902 * 12318305 = 11111111110"
"741: 819 * 12333469 = 10101111111"
"742: 9 * 12345679 = 111111111"
"743: 90 * 12345679 = 1111111110"
"744: 900 * 12345679 = 11111111100"
"745: 887 * 12401353 = 11000000111"
"746: 797 * 12548433 = 10001101101"
"747: 787 * 12846253 = 10110001111"
"748: 855 * 12994282 = 11110111110"
"749: 844 * 13045025 = 11010001100"
"750: 773 * 13080207 = 10111000011"
"751: 839 * 13230049 = 11100011111"
"752: 837 * 13274803 = 11111010111"
"753: 744 * 13440875 = 10000011000"
"754: 818 * 13447445 = 11000010010"
"755: 824 * 13470875 = 11100001000"
"756: 741 * 13643861 = 10110101001"
"757: 733 * 13644067 = 10001101111"
"758: 81 * 13717421 = 1111111101"
"759: 810 * 13717421 = 11111111010"
"760: 809 * 13721879 = 11101000111"
"761: 726 * 13775635 = 10001111010"
"762: 723 * 13832657 = 10001011011"
"763: 802 * 13853005 = 11110110010"
"764: 798 * 13922445 = 11110111110"
"765: 718 * 13941645 = 10010101110"
"766: 796 * 13944725 = 11100001100"
"767: 784 * 14030625 = 11000010000"
"768: 708 * 14138575 = 10010111100"
"769: 776 * 14304125 = 11100001000"
"770: 701 * 14422411 = 10110110111"
"771: 768 * 14453125 = 11100000000"
"772: 761 * 14469251 = 11011100011"
"773: 767 * 14473403 = 11101100101"
"774: 691 * 14486411 = 10010110001"
"775: 694 * 14554915 = 10101111010"
"776: 678 * 14898245 = 10101010110"
"777: 736 * 14959375 = 11010100000"
"778: 654 * 15305965 = 10010101110"
"779: 712 * 15463625 = 11010101000"
"780: 643 * 15569207 = 10011000101"
"781: 647 * 15625983 = 10110011001"
"782: 711 * 15627301 = 11111011011"
"783: 698 * 15760745 = 11001000010"
"784: 696 * 15819125 = 11010111000"
"785: 632 * 15838625 = 10010011000"
"786: 688 * 16004375 = 11011010000"
"787: 681 * 16169031 = 11011110111"
"788: 687 * 16171923 = 11110111101"
"789: 659 * 16859029 = 11110100111"
"790: 599 * 16863289 = 10101110111"
"791: 581 * 17383821 = 10100000001"
"792: 573 * 17452007 = 10000000011"
"793: 63 * 17635097 = 1111011111"
"794: 630 * 17635097 = 11110111110"
"795: 619 * 17786769 = 11010010011"
"796: 618 * 17962945 = 11101100010"
"797: 552 * 18116125 = 10000101000"
"798: 602 * 18289055 = 11010011110"
"799: 601 * 18304511 = 11001011111"
"800: 545 * 18367158 = 10010101110"
"801: 591 * 18629611 = 11010100101"
"802: 589 * 18694399 = 11011001011"
"803: 534 * 18934665 = 10111111110"
"804: 578 * 19033045 = 11001100010"
"805: 516 * 19399225 = 10010000100"
"806: 556 * 19803975 = 11011010100"
"807: 494 * 20242915 = 10000000010"
"808: 543 * 20460407 = 11110001001"
"809: 476 * 21031725 = 10011101100"
"810: 453 * 22077287 = 10001011011"
"811: 499 * 22244489 = 11100000011"
"812: 498 * 22311245 = 11111000010"
"813: 447 * 22371613 = 10000111011"
"814: 446 * 22446435 = 10011110010"
"815: 439 * 22781549 = 10001100011"
"816: 477 * 23293713 = 11111101101"
"817: 472 * 23328625 = 11011111000"
"818: 428 * 23366825 = 10001001100"
"819: 464 * 23730625 = 11011010000"
"820: 45 * 24691358 = 1111111110"
"821: 450 * 24691358 = 11111111100"
"822: 441 * 25195261 = 11111110101"
"823: 413 * 26879177 = 11101100101"
"824: 405 * 27434842 = 11111111010"
"825: 394 * 27944165 = 11010001010"
"826: 353 * 28615017 = 10101101001"
"827: 387 * 28710623 = 11111011101"
"828: 346 * 29219685 = 10110011010"
"829: 342 * 32485705 = 11110111110"
"830: 307 * 32576873 = 10001100011"
"831: 299 * 33478599 = 10010101101"
"832: 315 * 35270194 = 11110111110"
"833: 283 * 35371417 = 10010111011"
"834: 261 * 38739851 = 10111101111"
"835: 279 * 39824409 = 11111010111"
"836: 27 * 40781893 = 1101111111"
"837: 270 * 40781893 = 11011111110"
"838: 229 * 43668559 = 10000100011"
"839: 218 * 45917895 = 10010101110"
"840: 225 * 49382716 = 11111111100"
"841: 207 * 53193773 = 11011111011"
"842: 18 * 61728395 = 1111111110"
"843: 180 * 61728395 = 11111111100"
"844: 157 * 64338223 = 10101101011"
"845: 153 * 66085687 = 10111110111"
"846: 162 * 68587105 = 11111111010"
"847: 123 * 81309757 = 10001100111"
"848: 135 * 81563786 = 11011111110"
"849: 117 * 86334283 = 10101111111"
"850: 126 * 88175485 = 11110111110"
"851: 988 * 101214575 = 100000000100"
"852: 987 * 101419453 = 100101000111"
"853: 953 * 105037787 = 100101011011"
"854: 952 * 105158625 = 100111011000"
"855: 931 * 108486681 = 101001100011"
"856: 906 * 110386435 = 100010110110"
"857: 911 * 110867191 = 101000011001"
"858: 998 * 111222445 = 111000000110"
"859: 997 * 111444333 = 111110000001"
"860: 996 * 111556225 = 111110000100"
"861: 894 * 111858065 = 100001110110"
"862: 893 * 111983327 = 100001111011"
"863: 892 * 112232175 = 100111100100"
"864: 878 * 113907745 = 100011000110"
"865: 969 * 114655419 = 111101101011"
"866: 963 * 115275297 = 111010111011"
"867: 954 * 116468565 = 111111011010"
"868: 867 * 116505203 = 101010011001"
"869: 944 * 116643125 = 110111110000"
"870: 856 * 116834125 = 100010011000"
"871: 859 * 117707929 = 101111111011"
"872: 928 * 118653125 = 110110100000"
"873: 841 * 119036861 = 100110000101"
"874: 927 * 119850163 = 111101101101"
"875: 882 * 125976305 = 111111101010"
"876: 873 * 126015007 = 110011101111"
"877: 863 * 127463627 = 110001110101"
"878: 861 * 127770151 = 110010100011"
"879: 789 * 128137009 = 101100100101"
"880: 785 * 128676446 = 101011010110"
"881: 769 * 130053329 = 100011010001"
"882: 765 * 132171374 = 101111101110"
"883: 757 * 133554823 = 101101001011"
"884: 747 * 134016213 = 100110111111"
"885: 826 * 134395885 = 111011001010"
"886: 817 * 135875153 = 111010000001"
"887: 729 * 138698369 = 101111111001"
"888: 727 * 139064663 = 101100010001"
"889: 791 * 139078521 = 110011110111"
"890: 788 * 139720825 = 110100010100"
"891: 719 * 140472879 = 101000000001"
"892: 706 * 143075085 = 101011010010"
"893: 774 * 143553115 = 111110111010"
"894: 692 * 146098425 = 101100110100"
"895: 683 * 146413047 = 100000111101"
"896: 661 * 152798941 = 101000100001"
"897: 657 * 153898023 = 101111001111"
"898: 709 * 156574189 = 111011100001"
"899: 639 * 156668249 = 100111011111"
"900: 684 * 162428525 = 111101111100"
"901: 615 * 162619514 = 100011001110"
"902: 621 * 162642691 = 101001111111"
"903: 614 * 162884365 = 100011000110"
"904: 675 * 163127572 = 110111111100"
"905: 598 * 167392995 = 100101011010"
"906: 585 * 172668566 = 101011111110"
"907: 633 * 173777267 = 110001010011"
"908: 566 * 176857085 = 100101110110"
"909: 531 * 190228081 = 101011111011"
"910: 522 * 193699255 = 101111011110"
"911: 509 * 196662279 = 100101100011"
"912: 559 * 196799839 = 110011110001"
"913: 558 * 199122045 = 111110101110"
"914: 549 * 202205849 = 111011011101"
"915: 491 * 203871711 = 100101010101"
"916: 54 * 203909465 = 11011111110"
"917: 540 * 203909465 = 110111111100"
"918: 523 * 210537287 = 110111001101"
"919: 513 * 214622047 = 110101110111"
"920: 458 * 218342795 = 100001000110"
"921: 436 * 229589475 = 100101011100"
"922: 471 * 233781531 = 110111101101"
"923: 423 * 236669057 = 100111011111"
"924: 459 * 242050329 = 111101101011"
"925: 414 * 265968865 = 110111110110"
"926: 391 * 281613811 = 110111000101"
"927: 369 * 300816019 = 111001111011"
"928: 36 * 308641975 = 11111111100"
"929: 360 * 308641975 = 111111111000"
"930: 314 * 321691115 = 101011010110"
"931: 306 * 330428435 = 101111101110"
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?
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]],
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[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]],
[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]],
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[987, [86, 84882]],
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[989, [472, 466808]],
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[994, [23, 22862]],
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[998, [67, 66866]],
[999, [223112, 222888888]],
[1000, [2, 2000]]]
"1 * 1 = 1"
"2 * 5 = 10"
"3 * 37 = 111"
"4 * 25 = 100"
"5 * 2 = 10"
"6 * 185 = 1110"
"7 * 143 = 1001"
"8 * 125 = 1000"
"9 * 12345679 = 111111111"
"10 * 1 = 10"
"11 * 1 = 11"
"12 * 925 = 11100"
"13 * 77 = 1001"
"14 * 715 = 10010"
"15 * 74 = 1110"
"16 * 625 = 10000"
"17 * 653 = 11101"
"18 * 61728395 = 1111111110"
"19 * 579 = 11001"
"20 * 5 = 100"
"21 * 481 = 10101"
"22 * 5 = 110"
"23 * 4787 = 110101"
"24 * 4625 = 111000"
"25 * 4 = 100"
"26 * 385 = 10010"
"27 * 40781893 = 1101111111"
"28 * 3575 = 100100"
"29 * 37969 = 1101101"
"30 * 37 = 1110"
"31 * 3581 = 111011"
"32 * 3125 = 100000"
"33 * 3367 = 111111"
"34 * 3265 = 111010"
"35 * 286 = 10010"
"36 * 308641975 = 11111111100"
"37 * 3 = 111"
"38 * 2895 = 110010"
"39 * 259 = 10101"
"40 * 25 = 1000"
"41 * 271 = 11111"
"42 * 2405 = 101010"
"43 * 25607 = 1101101"
"44 * 25 = 1100"
"45 * 24691358 = 1111111110"
"46 * 23935 = 1101010"
"47 * 213 = 10011"
"48 * 23125 = 1110000"
"49 * 22449 = 1100001"
"50 * 2 = 100"
"51 * 1961 = 100011"
"52 * 1925 = 100100"
"53 * 1887 = 100011"
"54 * 203909465 = 11011111110"
"55 * 2 = 110"
"56 * 17875 = 1001000"
"57 * 193 = 11001"
"58 * 189845 = 11011010"
"59 * 186629 = 11011111"
"60 * 185 = 11100"
"61 * 1641 = 100101"
"62 * 17905 = 1110110"
"63 * 17635097 = 1111011111"
"64 * 15625 = 1000000"
"65 * 154 = 10010"
"66 * 16835 = 1111110"
"67 * 16433 = 1101011"
"68 * 16325 = 1110100"
"69 * 144929 = 10000101"
"70 * 143 = 10010"
"71 * 141 = 10011"
"72 has no partner below 1000000000"
"73 * 137 = 10001"
"74 * 15 = 1110"
"75 * 148 = 11100"
"76 * 14475 = 1100100"
"77 * 13 = 1001"
"78 * 1295 = 101010"
"79 * 126709 = 10010011"
"80 * 125 = 10000"
"81 * 13717421 = 1111111101"
"82 * 1355 = 111110"
"83 * 1217 = 101011"
"84 * 12025 = 1010100"
"85 * 1306 = 111010"
"86 * 128035 = 11011010"
"87 * 126553 = 11010111"
"88 * 125 = 11000"
"89 * 123709 = 11010101"
"90 * 12345679 = 1111111110"
"91 * 11 = 1001"
"92 * 119675 = 11010100"
"93 * 107527 = 10000011"
"94 * 1065 = 100110"
"95 * 1158 = 110010"
"96 * 115625 = 11100000"
"97 * 114433 = 11100001"
"98 * 112245 = 11000010"
[99 left as an exercise; it ain't pretty]