Polydivisible Number/Examples/3,608,528,850,368,400,786,036,725

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Theorem

The largest polydivisible number has $25$ digits:

$3 \, 608 \, 528 \, 850 \, 368 \, 400 \, 786 \, 036 \, 725$


Proof

\(\ds 3\) \(=\) \(\ds 1 \times 3\)
\(\ds 36\) \(=\) \(\ds 2 \times 18\)
\(\ds 360\) \(=\) \(\ds 3 \times 120\)
\(\ds 3608\) \(=\) \(\ds 4 \times 902\)
\(\ds 36 \, 085\) \(=\) \(\ds 5 \times 7217\)
\(\ds 360 \, 852\) \(=\) \(\ds 6 \times 60 \, 142\)
\(\ds 3 \, 608 \, 528\) \(=\) \(\ds 7 \times 515 \, 504\)
\(\ds 36 \, 085 \, 288\) \(=\) \(\ds 8 \times 4 \, 510 \, 661\)
\(\ds 360 \, 852 \, 885\) \(=\) \(\ds 9 \times 40 \, 094 \, 765\)
\(\ds 3 \, 608 \, 528 \, 850\) \(=\) \(\ds 10 \times 360 \, 852 \, 885\)
\(\ds 36 \, 085 \, 288 \, 503\) \(=\) \(\ds 11 \times 3 \, 280 \, 480 \, 773\)
\(\ds 360 \, 852 \, 885 \, 036\) \(=\) \(\ds 12 \times 30 \, 071 \, 073 \, 753\)
\(\ds 3 \, 608 \, 528 \, 850 \, 368\) \(=\) \(\ds 13 \times 277 \, 579 \, 142 \, 336\)
\(\ds 36 \, 085 \, 288 \, 503 \, 684\) \(=\) \(\ds 14 \times 2 \, 577 \, 520 \, 607 \, 406\)
\(\ds 360 \, 852 \, 885 \, 036 \, 840\) \(=\) \(\ds 15 \times 24 \, 056 \, 859 \, 002 \, 456\)
\(\ds 3 \, 608 \, 528 \, 850 \, 368 \, 400\) \(=\) \(\ds 16 \times 225 \, 533 \, 053 \, 148 \, 025\)
\(\ds 36 \, 085 \, 288 \, 503 \, 684 \, 007\) \(=\) \(\ds 17 \times 2 \, 122 \, 664 \, 029 \, 628 \, 471\)
\(\ds 360 \, 852 \, 885 \, 036 \, 840 \, 078\) \(=\) \(\ds 18 \times 20 \, 047 \, 382 \, 502 \, 046 \, 671\)
\(\ds 3 \, 608 \, 528 \, 850 \, 368 \, 400 \, 786\) \(=\) \(\ds 19 \times 189 \, 922 \, 571 \, 072 \, 021 \, 094\)
\(\ds 36 \, 085 \, 288 \, 503 \, 684 \, 007 \, 860\) \(=\) \(\ds 20 \times 1 \, 804 \, 264 \, 425 \, 184 \, 200 \, 393\)
\(\ds 360 \, 852 \, 885 \, 036 \, 840 \, 078 \, 603\) \(=\) \(\ds 21 \times 17 \, 183 \, 470 \, 716 \, 040 \, 003 \, 743\)
\(\ds 3 \, 608 \, 528 \, 850 \, 368 \, 400 \, 786 \, 036\) \(=\) \(\ds 22 \times 164 \, 024 \, 038 \, 653 \, 109 \, 126 \, 638\)
\(\ds 36 \, 085 \, 288 \, 503 \, 684 \, 007 \, 860 \, 367\) \(=\) \(\ds 23 \times 1 \, 568 \, 925 \, 587 \, 116 \, 695 \, 993 \, 929\)
\(\ds 360 \, 852 \, 885 \, 036 \, 840 \, 078 \, 603 \, 672\) \(=\) \(\ds 24 \times 15 \, 035 \, 536 \, 876 \, 535 \, 003 \, 275 \, 153\)
\(\ds 3 \, 608 \, 528 \, 850 \, 368 \, 400 \, 786 \, 036 \, 725\) \(=\) \(\ds 25 \times 144 \, 341 \, 154 \, 014 \, 736 \, 031 \, 441 \, 469\)

From No Polydivisible Number with 26 Digits Exists, there are no polydivisible numbers with more digits.


This theorem requires a proof.
In particular: It remains to be shown there is no larger one with $25$ digits.
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Sources

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