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

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

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



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Sources