Pandigital Numbers Divisible by All Integers up to 18

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Theorem

The following pandigital integers are divisible by all the positive integers up to $18$:

$2 \, 438 \, 195 \, 760$
$3 \, 785 \, 942 \, 160$
$4 \, 753 \, 869 \, 120$
$4 \, 876 \, 391 \, 520$


Proof

\(\ds 2 \, 438 \, 195 \, 760\) \(=\) \(\ds 2 \times 1 \, 219 \, 097 \, 880\)
\(\ds \) \(=\) \(\ds 3 \times 812 \, 731 \, 920\)
\(\ds \) \(=\) \(\ds 4 \times 609 \, 548 \, 940\)
\(\ds \) \(=\) \(\ds 5 \times 487 \, 639 \, 152\)
\(\ds \) \(=\) \(\ds 6 \times 406 \, 365 \, 960\)
\(\ds \) \(=\) \(\ds 7 \times 348 \, 313 \, 680\)
\(\ds \) \(=\) \(\ds 8 \times 304 \, 774 \, 470\)
\(\ds \) \(=\) \(\ds 9 \times 270 \, 910 \, 640\)
\(\ds \) \(=\) \(\ds 10 \times 243 \, 819 \, 576\)
\(\ds \) \(=\) \(\ds 11 \times 221 \, 654 \, 160\)
\(\ds \) \(=\) \(\ds 12 \times 203 \, 182 \, 980\)
\(\ds \) \(=\) \(\ds 13 \times 187 \, 553 \, 520\)
\(\ds \) \(=\) \(\ds 14 \times 174 \, 156 \, 840\)
\(\ds \) \(=\) \(\ds 15 \times 162 \, 546 \, 384\)
\(\ds \) \(=\) \(\ds 16 \times 152 \, 387 \, 235\)
\(\ds \) \(=\) \(\ds 17 \times 143 \, 423 \, 280\)
\(\ds \) \(=\) \(\ds 18 \times 135 \, 553 \, 320\)


\(\ds 3 \, 785 \, 942 \, 160\) \(=\) \(\ds 2 \times 1 \, 892 \, 971 \, 080\)
\(\ds \) \(=\) \(\ds 3 \times 1 \, 261 \, 980 \, 720\)
\(\ds \) \(=\) \(\ds 4 \times 946 \, 485 \, 540\)
\(\ds \) \(=\) \(\ds 5 \times 757 \, 188 \, 432\)
\(\ds \) \(=\) \(\ds 6 \times 630 \, 990 \, 360\)
\(\ds \) \(=\) \(\ds 7 \times 540 \, 848 \, 880\)
\(\ds \) \(=\) \(\ds 8 \times 473 \, 242 \, 770\)
\(\ds \) \(=\) \(\ds 9 \times 420 \, 660 \, 240\)
\(\ds \) \(=\) \(\ds 10 \times 378 \, 594 \, 216\)
\(\ds \) \(=\) \(\ds 11 \times 344 \, 176 \, 560\)
\(\ds \) \(=\) \(\ds 12 \times 315 \, 495 \, 180\)
\(\ds \) \(=\) \(\ds 13 \times 291 \, 226 \, 320\)
\(\ds \) \(=\) \(\ds 14 \times 270 \, 424 \, 440\)
\(\ds \) \(=\) \(\ds 15 \times 252 \, 396 \, 144\)
\(\ds \) \(=\) \(\ds 16 \times 236 \, 621 \, 385\)
\(\ds \) \(=\) \(\ds 17 \times 222 \, 702 \, 480\)
\(\ds \) \(=\) \(\ds 18 \times 210 \, 330 \, 120\)


\(\ds 4 \, 753 \, 869 \, 120\) \(=\) \(\ds 2 \times 2 \, 376 \, 934 \, 560\)
\(\ds \) \(=\) \(\ds 3 \times 1 \, 584 \, 623 \, 040\)
\(\ds \) \(=\) \(\ds 4 \times 1 \, 188 \, 467 \, 280\)
\(\ds \) \(=\) \(\ds 5 \times 950 \, 773 \, 824\)
\(\ds \) \(=\) \(\ds 6 \times 792 \, 311 \, 520\)
\(\ds \) \(=\) \(\ds 7 \times 679 \, 124 \, 160\)
\(\ds \) \(=\) \(\ds 8 \times 594 \, 233 \, 640\)
\(\ds \) \(=\) \(\ds 9 \times 528 \, 207 \, 680\)
\(\ds \) \(=\) \(\ds 10 \times 475 \, 386 \, 912\)
\(\ds \) \(=\) \(\ds 11 \times 432 \, 169 \, 920\)
\(\ds \) \(=\) \(\ds 12 \times 396 \, 155 \, 760\)
\(\ds \) \(=\) \(\ds 13 \times 365 \, 682 \, 240\)
\(\ds \) \(=\) \(\ds 14 \times 339 \, 562 \, 080\)
\(\ds \) \(=\) \(\ds 15 \times 316 \, 924 \, 608\)
\(\ds \) \(=\) \(\ds 16 \times 297 \, 116 \, 820\)
\(\ds \) \(=\) \(\ds 17 \times 279 \, 639 \, 360\)
\(\ds \) \(=\) \(\ds 18 \times 264 \, 103 \, 840\)


\(\ds 4 \, 876 \, 391 \, 520\) \(=\) \(\ds 2 \times 2 \, 438 \, 195 \, 760\) see above
\(\ds \) \(=\) \(\ds 3 \times 1 \, 625 \, 463 \, 840\)
\(\ds \) \(=\) \(\ds 4 \times 1 \, 210 \, 097 \, 880\)
\(\ds \) \(=\) \(\ds 5 \times 975 \, 278 \, 304\)
\(\ds \) \(=\) \(\ds 6 \times 812 \, 731 \, 920\)
\(\ds \) \(=\) \(\ds 7 \times 696 \, 627 \, 360\)
\(\ds \) \(=\) \(\ds 8 \times 609 \, 548 \, 940\)
\(\ds \) \(=\) \(\ds 9 \times 541 \, 821 \, 280\)
\(\ds \) \(=\) \(\ds 10 \times 487 \, 639 \, 152\)
\(\ds \) \(=\) \(\ds 11 \times 443 \, 308 \, 320\)
\(\ds \) \(=\) \(\ds 12 \times 406 \, 365 \, 960\)
\(\ds \) \(=\) \(\ds 13 \times 375 \, 107 \, 040\)
\(\ds \) \(=\) \(\ds 14 \times 348 \, 313 \, 680\)
\(\ds \) \(=\) \(\ds 15 \times 325 \, 092 \, 768\)
\(\ds \) \(=\) \(\ds 16 \times 304 \, 774 \, 470\)
\(\ds \) \(=\) \(\ds 17 \times 286 \, 846 \, 560\)
\(\ds \) \(=\) \(\ds 18 \times 270 \, 910 \, 640\)


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Sources

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