3-digit Num3ers & Permutations

I’m interested in 3-digit numbers with 3 different digits are chosen from among 1 – 9. And I’ll exclude the prime numbers.

The 3-digit prime numbers : (List #1)

101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,181,191,193,197,199,
211,223,227,229,233,239,241,251,257,263,269,271,277,281,283,293,307,311,313,317,331,
337,347,349,353,359,367,373,379,383,389,397,401,409,419,421,431,433,439,443,449,457,
461,463,467,479,487,491,499,503,509,521,523,541,547,557,563,569,571,577,587,593,599,
601,607,613,617,619,631,641,643,647,653,659,661,673,677,683,691,701,709,719,727,733,
739,743,751,757,761,769,773,787,797,809,811,821,823,827,829,839,853,857,859,863,877,
881,883,887,907,911,919,929,937,941,947,953,967,971,977,983,991,997

The list of 3-digit numbers : (List #2)

100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,
120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,
140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,
160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,
180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,
200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,
220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,
240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,
260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,
280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,
300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,
320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,
340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,
360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,
380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,
400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,
420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,
440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,
460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,
480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,
500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,
520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,
540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,
560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,
580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,
600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,
620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,
640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,
660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,
680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,
700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,
720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,
740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,
760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,
780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,
800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,
820,821,822,823,824,825,826,827, 828,829,830,831,832,833,834,835,836,837,838,839,
840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,
860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,
880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,896,897,898,899,
900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,
920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,
940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,
960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,
980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999

So we have now the list of numbers (L #3) :

123, 124, 125, 126, 128, 129, 132, 134, 135, 136, 138, 142, 143, 145, 146,
147, 148, 152, 153, 154, 156, 158, 159, 162, 164, 165, 168, 169, 172, 174,
175, 176, 178, 182, 183, 184, 185, 186, 187, 189, 192, 194, 195, 196, 198, 213,
214, 215, 216, 217, 218, 219, 231, 234, 235, 236, 237, 238, 243, 245, 246,
247, 248, 249, 253, 254, 256, 258, 259, 261, 264, 265, 267, 268, 273, 274,
275, 276, 278, 279, 284, 285, 286, 287, 289, 291, 294, 295, 296, 297, 298,
312, 314, 315, 316, 318, 319, 321, 324, 325, 326, 327, 328, 329, 341, 342,
345, 346, 348, 351, 352, 354, 356, 357, 358, 361, 362, 364, 365, 367, 368,
369, 371, 372, 374, 375, 376, 378, 381, 382, 384, 385, 386, 387, 391, 392,
394, 395, 396, 398, 412, 413, 415, 416, 417, 418, 423, 425, 426, 427, 428,
429,  432, 435, 436, 437, 438, 451, 452, 453, 456, 458, 459, 462, 465,
468, 469, 471, 472, 473, 475, 476, 478, 481, 482, 483, 485, 486, 487, 489,
492, 493, 495, 496, 497, 498, 512, 513, 514, 516, 517, 518, 519, 524, 526,
527, 528, 529, 531, 532, 534, 536, 537, 538, 539, 542, 543, 546, 548, 549,
561, 562, 564, 567, 568, 572, 573, 574, 576, 578, 579, 581, 582, 583, 584,
586, 589, 591, 592, 594, 596, 597, 598, 612, 614, 615, 618, 621, 623, 624,
627, 628, 629, 632, 634, 635, 637, 638, 639, 642, 645, 648, 649, 651, 652,
654, 657, 658, 671, 672, 674, 675, 678, 679, 681, 682, 684, 685, 687, 689,
692, 693, 694, 695, 697, 698, 712, 713, 714, 715, 716, 721, 723, 724, 725, 726,
728, 729, 731, 732, 734, 735, 736, 738, 741, 742, 745, 746, 748, 749, 752,
753, 754, 756, 758, 759, 762, 763, 764, 765, 768, 781, 782, 783, 784, 785,
786, 789, 791, 792, 793, 794, 795, 796, 798, 812, 813, 814, 815, 816, 817,
819, 824, 825, 826, 831, 832, 834, 835, 836, 837, 841, 842, 843, 845, 846,
847, 849, 851, 852, 854, 856, 861, 862, 864, 865, 867, 869, 871, 872, 873,
874, 875, 876, 879, 891, 892, 893, 894, 895, 896, 897, 912, 913, 914, 915,
916, 917, 918, 921, 923, 924, 925, 926, 927, 928, 931, 932, 934, 935, 936,
938, 921, 923, 924, 925, 926, 927, 928, 931, 932, 934, 935, 936, 938, 942,
943, 945, 946, 948, 951, 952, 954, 956, 957, 958, 961, 962, 963, 964, 965,
968, 972, 973, 974, 975, 976, 978, 981, 982, 984, 985, 986, 987.

To find which numbers can be arranged in each of the six possible orders, and which 3-digit numbers is the product of exactly two primes

From the list (List #3), we have:

> 123, 132, 213, 231, 312, 321     N/A
123 = 3 * 41,     132 = 2 * 2 * 3 * 11,     213 = 3 * 71
231 = 3 * 7 * 11,     312 = 2 * 2 * 2 * 3 * 13,     321 = 3 * 107

> 124, 142, 214, ___ , 412, ___ .        N/A
> 125, 152, 215, ___ , 512, ___ .        N/A

> 126, 162, 216, 261, 612, 621     N/A
126 = 2*3*3*7,    162 = 2*3*3*3*3,    216 = 2*2*2*3*3*3
261 = 3*3*29,    612 = 2*2*3*3*17,    621 = 3*3*3*23

> ___ , 172, 217, ___ , 712, 721        N/A
> 128, 182, 218, ___ , 812, ___ .     N/A

> 129, 192, 219, 291, 912, 921     N/A
129 = 3*43,    192 = 2*2*2*2*2*2*3,    219 = 3*73
291 = 3*97,    912 = 2*2*2*2*3*19,    921 = 3*307

> 134, 143, 314, 341, 413, ___  N/A

> 135, 153, 315, 351, 513, 531      N/A
135 = 3*3*3*5,    153 = 3*3*17,    315 = 3*3*5*7
351 = 3*3*3*13,    513 = 3*3*3*19,    531 = 3*3*59

> 136, ___ , 316, 361, ___ , ___ .        N/A
> 138, 183, 318, 381, 813, 831      N/A
138 = 2*3*23,    183 = 3*61,    318 = 2*3*53
381 = 3*127,    813 = 3*271,    831 = 3*277

___ , ___ , 319, 391, 913, 931     N/A
> 145, 154, 415, 451, 514, ___ .        N/A
> 146, 164, 416, ___ , 614, ___ .        N/A
> 147, 174, 417, 471, 714, 741      N/A
147 = 3*7*7,    174 = 2*3*29,    417 = 3*139
471 = 3*157,    714 = 2*3*7*17,    741 = 3*13*19

> 148, 184, 418, 481, 814, 841      N/A
148 = 2*2*37,    184 = 2*2*2*23,    418 = 2*11*19
481 = 13*37,    814 = 2*11*37,    841 = 29*29

> 156, 165, 516, 561, 615, 651      N/A
156 = 2*2*3*13,    165 = 3*5*11,    516 = 2*2*3*43
561 = 3*11*17,    615 = 3*5*41,    651 = 3*7*31

> ___ , 175, 517, ___ , 715, ___      N/A
> 158, 185, 518, 581, 815, 851      N/A
158 = 2*79,    185 = 5*37,    518 = 2*7*37
581 = 7*83,    815 = 5*163,    851 = 23*37

> 159, 195, 519, 591, 915, 951      N/A
159 = 3*53,    195 = 3*5*13,    519 = 3*173
591 = 3*197,    915 = 3*5*61,    951 = 3*317

> ___ , 176, ___ , 671, 716, ___        N/A
168, 186, 618, 681, 816, 861      N/A
168 = 2*2*2*3*7,    186 = 2*3*31,    618 = 2*3*103
681 = 3*227,    816 = 2*2*2*2*3*17,    861 = 3*7*41

> 169, 196, ___ , ___ , 916, 961        N/A

> 178, 187, 718, 781, 817, 871
178 = 2*89,    187 = 11*17,    718 = 2*359
781 = 11*71,    817 = 19*43,    871 = 13*67

> ___ , ___ , ___ , 791, 917      N/A

> 189, 198, 819, 891, 918, 981      N/A
189 = 3*3*3*7,    198 = 2*3*3*11,    819 = 3*3*7*13
891 = 3*3*3*3*11,    918 = 2*3*3*3*17,    981 = 3*3*109

> ___ , 194, ___ , ___ , 914, ___     N/A
> 234, 243, 324, 342, 423, 432      N/A
234 = 2*3*3*13,    243 = 3*3*3*3*3,    324 = 2*2*3*3*3*3
342 = 2*3*3*19,    423 = 3*3*47,    432 = 2*2*2*2*3*3*3

> 235, 253, 325, 352, ___ , 532     N/A
> 236, ___ , 326, 362, 623, 632     N/A

> 237, 273, 327, 372, 723, 732      N/A
237 = 3*79,    273 = 3*7*13,    327 = 3*109
372 = 2*2*3*31,    723 = 3*241,    732 = 2*2*3*61

> 238, ___ , 328, 382, ___ , 832     N/A
> ___ , ___ , 329, 392, 923, 932     N/A

> 245, 254, 425, 452, 524, 542      N/A
245 = 5*7*7,    254 = 2*127,    425 = 5*5*17
452 = 2*2*113,    524 = 2*2*131,    542 = 2*271

> 246, 264, 426, 462, 624, 642      N/A
246 = 2*3*41,    264 = 2*2*2*3*11,    426 = 2*3*71
462 = 2*3*7*11,    624 = 2*2*2*2*3*13,    642 = 2*3*107

> 247, 274, 427, 472, 724, 742      N/A
247 = 13*19,    274 = 2*137,    427 = 7*61
472 = 2*2*2*59,    724 = 2*2*181,    742 = 2*7*53

> 248, 284, 428, 482, 824, 842      N/A
248 = 2*2*2*31,    284 = 2*2*71,    428 = 2*2*107
482 = 2*241,    824 = 2*2*2*103,    842 = 2*421

> 249, 294, 429, 492, 924, 942      N/A
249 = 3*83,    294 = 2*3*7*7,    429 = 3*11*13
492 = 2*2*3*41,    924 = 2*2*3*7*11,    942 = 2*3*157

> 256, 265, 526, 562, 625, 652      N/A
256 = 2*2*2*2*2*2*2*2,    265 = 5*53,    526 = 2*263
562 = 2*281,    625 = 5*5*5*5,    652 = 2*2*163

> ___ , 275, 527, 572, 725, 752          N/A

> 258, 285, 528, 582, 825, 852      N/A
258 = 2*3*43,    285 = 3*5*19,    528 = 2*2*2*2*3*11
582 = 2*3*97,    825 = 3*5*5*11,    852 = 2*2*3*71

> 259, 295, 529, 592, 925, 952      N/A
259 = 7*37,    295 = 5*59,    529 = 23*23
592 = 2*2*2*2*37,    925 = 5*5*37,    952 = 2*2*2*7*17

> 267, 276, 627, 672, 726, 762      N/A
267 = 3*89,    276 = 2*2*3*23,    627 = 3*11*19
672 = 2*2*2*2*2*3*7,    726 = 2*3*11*11,    762 = 2*3*127

> 268, 286, 628, 682, 826, 862 nbsp;    N/A
268 = 2*2*67,    286 = 2*11*13,    628 = 2*2*157
682 = 2*11*31,    826 = 2*7*59,    862 = 2*431

> ___ , 296, 629, 692, 926, 962         N/A

> 278, 287, 728, 782, ___ , 872         N/A

> 279, 297, 729, 792, 927, 972      N/A
279 = 3*3*31,    297 = 3*3*3*11,    729 = 3*3*3*3*3*3
792 = 2*2*2*3*3*11,    927 = 3*3*103,    972 = 2*2*3*3*3*3*3

> 289, 298, ___ , 892, 928, 982         N/A

> ___ , ___ , ___ , 371, 713, 731         N/A

> 345, 354, 435, 453, 534, 543      N/A
345 = 3*5*23,    354 = 2*3*59,    435 = 3*5*29
453 = 3*151,    534 = 2*3*89,    543 = 3*181

> 346, 364, 436, ___ , 634, ___         N/A

> ___ , 374, 437, 473, 734, ___         N/A

> 348, 384, 438, 483, 834, 843      N/A
348 = 2*2*3*29,    384 = 2*2*2*2*2*2*2*3,    438 = 2*3*73
483 = 3*7*23,    834 = 2*3*139,    843 = 3*281

> ___ , 394, ___ , 493, 934, 943      N/A

> 356, 365, 536, ___ , 635, ___            N/A

> 357, 375, 537, 573, 735, 753      N/A
357 = 3*7*17,    375 = 3*5*5*5,    537 = 3*179
573 = 3*191,    735 = 3*5*7*7,    753 = 3*251

> 358, 385, 538, 583, 835, ___            N/A

> ___ , 395, 539, ___ , 935, ___            N/A

> 367, 376, 637, ___ , 736, 763            N/A
> 368, 386, 638, ___ , 836, ___            N/A

> 369, 396, 639, 693, 936, 963      N/A
369 = 3*3*41,    396 = 2*2*3*3*11,    639 = 3*3*71
693 = 3*3*7*11,    936 = 2*2*2*3*3*13,    963 = 3*3*107

> 378, 387, 738, 783, 837, 873      N/A
378 = 2*3*3*3*7,    387 = 3*3*43,    738 = 2*3*3*41
783 = 3*3*3*29,    837 = 3*3*3*31,    873 = 3*3*97

> ___ , ___ , ___ , 793, ___ , 973         N/A
> ___ , 398, ___ , 893, 938, ___            N/A

> 456, 465, 546, 564, 645, 654      N/A
456 = 2*2*2*3*19,    465 = 3*5*31,    546 = 2*3*7*13
564 = 2*2*3*47,    645 = 3*5*43,    654 = 2*3*109

> ___ , 475, ___ , 574, 745, 754             N/A

> 458, 485, 548, 584, 845, 854           N/A
458 = 2*229,    485 = 5*97,    548 = 2*2*137
584 = 2*2*2*73,    845 = 5*13*13,    854 = 2*7*61

> 459, 495, 549, 594, 945, 954          N/A
459 = 3*3*3*17,    495 = 3*3*5*11,    549 = 3*3*61
594 = 2*3*3*3*11,    945 = 3*3*3*5*7,    954 = 2*3*3*53

> ___ , 476, ___ , 674, 746, 764            N/A

> 468, 486, 648, 684, 846, 864         N/A
468 = 2*2*3*3*13,    486 = 2*3*3*3*3*3,    648 = 2*2*2*3*3*3*3
684 = 2*2*3*3*19,    846 = 2*3*3*47,    864 = 2*2*2*2*2*3*3*3

> 469, 496, 649, 694, 946, 964         N/A
469 = 7*67,    496 = 2*2*2*2*31,    649 = 11*59
694 = 2*347,    946 = 2*11*43,    964 = 2*2*241

> 478, ___ , 748, 784, 847, 874         N/A
> ___ , 497, 749, 794, ___ , 974            N/A

> 489, 498, 849, 894, 948, 984            N/A
489 = 3*163,    498 = 2*3*83,    849 = 3*283
894 = 2*3*149,    948 = 2*2*3*79,    984 = 2*2*2*3*41

> 567, 576, 657, 675, 756, 765            N/A
567 = 3*3*3*3*7,    576 = 2*2*2*2*2*2*3*3,    657 = 3*3*73
675 = 3*3*3*5*5,    756 = 2*2*3*3*3*7,    765 = 3*3*5*17

> 568, 586, 658, 685, 856, 865            N/A
568 = 2*2*2*71,    586 = 2*293,    658 = 2*7*47
685 = 5*137,    856 = 2*2*2*107,    865 = 5*173

> ___ , 596, 659, 695, 956, 965         N/A
> 578, ___ , 758, 785, ___ , 875         N/A
> 579, 597, 759, 795, 957, 975         N/A
579 = 3*193,    597 = 3*199,    759 = 3*11*23
795 = 3*5*53,    957 = 3*11*29,    975 = 3*5*5*13

> 589, 598, ___ , 895, 958, 985         N/A

> 678, 687, 768, 786, 867, 876          N/A
678 = 2*3*113,    687 = 3*229,    768 = 2*2*2*2*2*2*2*2*3
786 = 2*3*131,    867 = 3*17*17,    876 = 2*2*3*73

> 679, 697, ___ , 796, ___ , 976         N/A
> 689, 698, 869, 896, 968, 986          N/A
689 = 13*53,    698 = 2*349,    869 = 11*79
896 = 2*2*2*2*2*2*2*7,    968 = 2*2*2*11*11,    986 = 2*17*29

> 789, 798, 879, 897, 978, 987          N/A
789 = 3 * 263,    798 = 2 * 3 * 7 * 19,    879 = 3 * 293
897 = 3 * 13 * 23,    978 = 2 * 3 * 163,    987 = 3 * 7 * 47

The only solution is :

178, 187, 718, 781, 817, 871
178 = 2 * 89
187 = 11 * 17
718 = 2 * 359
781 = 11 * 71
817 = 19 * 43
871 = 13 * 67

About these ads