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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_
# 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
@Vaguery
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Vaguery commented Feb 19, 2014

"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]

@Vaguery
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Vaguery commented Feb 20, 2014

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.

@Vaguery
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Vaguery commented Feb 21, 2014

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"

@Vaguery
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Vaguery commented Feb 21, 2014

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?

@Vaguery
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Vaguery commented Feb 22, 2019

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]],
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 [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]]]

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Vaguery commented Feb 22, 2019

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]]]

@Vaguery
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Vaguery commented Feb 22, 2019

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]],
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