Problems & Puzzles: Puzzles

Puzzle 1161 8757193191
G. L. Honaker, Jr. sent the following nice puzzle.

Just as a side note to Puzzle 1158,
here's a curio I thought you'd enjoy
seeing:

https://t5k.org/curios/page.php?number_id=85

8757193191
This number is a composite.

+ This number is the provably largest number with the property that the first N digits are divisible by the Nth prime, for all appropriate N. [Keith]

Q. Can you confirm it or find a larger example

From 2-8 Feb, 2024 contributions came from Giorgos Kalogeropoulos, Michael Branicky, Ivan Ianakev, Gennady Gusev, Paul Cleary, Emmanuel Vantieghem

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Giorgos wrote:

This number is the largest possible as it is clearly stated in  A225613
 Here are all the possible numbers for each N
 

 
N=1->{2,4,6,8}
N=2->{21,24,27,42,45,48,60,63,66,69,81,84,87}
N=3->{210,215,240,245,270,275,420,425,450,455,480,485,600,605,630,635,660,665,690,695,810,815,840,845,870,875}
N=4->{2100,2107,2156,2401,2408,2450,2457,2702,2709,2751,2758,4200,4207,4256,4501,4508,4550,4557,4802,4809,
4851,4858,6006,6055,6300,6307,6356,6601,6608,6650,6657,6902,6909,6951,6958,8106,8155,8400,8407,8456,8701,8708,8750,8757}
N=5->{21076,21560,24013,24508,24574,27027,27093,27511,27588,42009,42075,45012,45089,45507,45573,48026,48092,48510,48587,
60060,60555,63008,63074,63569,66011,66088,66506,66572,69025,69091,69586,81554,84007,84073,84568,87010,87087,87505,87571}
N=6->{210769,215605,240136,245089,270270,270933,275119,275886,420095,420758,450125,450892,455078,480922,485108,485875,
600600,605553,630084,630747,660114,660881,665067,690911,695864,815542,840073,840736,845689,870103,870870,875056,875719}
N=7->{2107694,2156059,2401369,2450890,2751195,2758862,4200955,4207585,4508927,4550781,4851086,4858753,6055536,6300846,
6307476,6608818,6650672,6958644,8155427,8400737,8407367,8708709,8750563,8757193}
N=8->{21076947,21560592,27588627,42009551,45507812,63008465,66506726,81554270,84007379,84073670,87571931}
N=9->{210769470,630084655,665067264,875719319}
N=10->{6300846559,8757193191}
N=11->{}

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Michael wrote:

There are exactly 200 positive numbers satisfying the condition and 8757193191 is the largest.

 
Listed by number of digits, the terms are as follows:
1: 2, 4, 6, 8 2: 21, 24, 27, 42, 45, 48, 60, 63, 66, 69, 81, 84, 87 3: 210, 215, 240, 245, 270, 275, 420, 425, 450, 455, 480, 485, 600, 605, 630, 635, 660, 665, 690, 695, 810, 815, 840, 845, 870, 875 4: 2100, 2107, 2156, 2401, 2408, 2450, 2457, 2702, 2709, 2751, 2758, 4200, 4207, 4256, 4501, 4508, 4550, 4557, 4802, 4809, 4851, 4858, 6006, 6055, 6300, 6307, 6356, 6601, 6608, 6650, 6657, 6902, 6909, 6951, 6958, 8106, 8155, 8400, 8407, 8456, 8701, 8708, 8750, 8757 5: 21076, 21560, 24013, 24508, 24574, 27027, 27093, 27511, 27588, 42009, 42075, 45012, 45089, 45507, 45573, 48026, 48092, 48510, 48587, 60060, 60555, 63008, 63074, 63569, 66011, 66088, 66506, 66572, 69025, 69091, 69586, 81554, 84007, 84073, 84568, 87010, 87087, 87505, 87571 6: 210769, 215605, 240136, 245089, 270270, 270933, 275119, 275886, 420095, 420758, 450125, 450892, 455078, 480922, 485108, 485875, 600600, 605553, 630084, 630747, 660114, 660881, 665067, 690911, 695864, 815542, 840073, 840736, 845689, 870103, 870870, 875056, 875719 7: 2107694, 2156059, 2401369, 2450890, 2751195, 2758862, 4200955, 4207585, 4508927, 4550781, 4851086, 4858753, 6055536, 6300846, 6307476, 6608818, 6650672, 6958644, 8155427, 8400737, 8407367, 8708709, 8750563, 8757193 8: 21076947, 21560592, 27588627, 42009551, 45507812, 63008465, 66506726, 81554270, 84007379, 84073670, 87571931 9: 210769470, 630084655, 665067264, 875719319 10: 6300846559, 8757193191
 

 
These were found by performing breadth-first search on feasible candidates, where depth is the number of decimal digits.
Neither of the 10-digit solutions can be extended, so the search ends.

 
The program is linked here:

 
After solving, a search of the OEIS shows the numbers are captured in https://oeis.org/A079206

***

Ivan wrote:

The definition clearly says this is the PROVABLY largest number. The proof is based on the observation that
the only 10-digit such number is 8757193191 and there are no numbers in [87571931910, 87571931919] divisible by prime(11).

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Gennady wrote:

There are only two 10-digit numbers with this property:
                           6300846559
                           8757193191
And 8757193191 is the largest.
There are no such 11-digit numbers.

***

Paul wrote:

I solved this problem on 15th Sept 2013 and can confirm that  8757193191 is the maximum number.
p.s. if i remember correctly this was a puzzle set by you on the digits page by F Rubin.:

Divisible by Primes (contributed by Carlos Rivera)
Find the largest integer A such that the first K digits of A form an integer divisible by PK the K-th prime for K=1,2,3,...,N where N is the number of digits in A. Similarly, find the largest integer where the last K digits are divisible by PK.

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Emmanuel wrote:

Let  A(n)  be the list of all  n-digit numbers whose k first digits are divisible by p(k) for all  k <= n .
Then we have :
A(1) = {2, 4, 6, 8};
A(2) = {21, 24, 27, 42, 45, 48, 60, 63, 66, 69, 81, 84, 87}
A(3) = {210, 215, 240, 245, 270, 275, 420, 425, 450, 455, 480, 485, 600, 605, 630, 635, 660, 665, 690, 695, 810, 815, 840, 845, 870, 875}
A(4) = {2100, 2107, 2156, 2401, 2408, 2450, 2457, 2702, 2709, 2751, 2758, 4200, 4207, 4256, 4501, 4508, 4550, 4557, 4802, 4809, 4851, 4858, 6006, 6055, 6300, 6307, 6356, 6601, 6608, 6650, 6657, 6902, 6909, 6951, 6958, 8106, 8155, 8400, 8407, 8456, 8701, 8708, 8750, 8757}
A(5) = {21076, 21560, 24013, 24508, 24574, 27027, 27093, 27511, 27588, 42009, 42075, 45012, 45089, 45507, 45573, 48026, 48092, 48510, 48587, 60060, 60555, 63008, 63074, 63569, 66011, 66088, 66506, 66572, 69025, 69091, 69586, 81554, 84007, 84073, 84568, 87010, 87087, 87505, 87571}
A(6) = {210769, 215605, 240136, 245089, 270270, 270933, 275119, 275886, 420095, 420758, 450125, 450892, 455078, 480922, 485108, 485875, 600600, 605553, 630084, 630747, 660114, 660881, 665067, 690911, 695864, 815542, 840073, 840736, 845689, 870103, 870870, 875056, 875719}
A(7) = {2107694, 2156059, 2401369, 2450890, 2751195, 2758862, 4200955, 4207585, 4508927, 4550781, 4851086, 4858753, 6055536, 6300846, 6307476, 6608818, 6650672, 6958644, 8155427, 8400737, 8407367, 8708709, 8750563, 8757193}
A(8) = {21076947, 21560592, 27588627, 42009551, 45507812, 63008465, 66506726, 81554270, 84007379, 84073670, 87571931}
A(9) = {210769470, 630084655, 665067264, 875719319}
A(10) ={6300846559, 8757193191}
A(11) = {}
This proves that  8757193191  is indeed the biggest possible.
As a by-product : 6300846559  is the smallest such ten-digit integer

 

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