Polydivisible number: Difference between revisions

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{{refimprove|date=October 2018}}

In [[mathematics]] a '''polydivisible number''' (or '''magic number''') is a [[natural number|number]] in a given [[number base]] with [[numerical digit|digits]] ''abcde...'' that has the following properties :

# Its first digit ''a'' is not 0.

In mathematics a '''polydivisible number''' (or '''magic number''') is a number with digits ''abcdef...'' that has the following properties :

# ItsThe number formed by its first digit two digits ''aab''  is nota multiple of 02.

# The number formed by its first twothree digits  ''ababc''  is a multiple of 23.

# The number formed by its first threefour digits  ''abcabcd''  is a multiple of 34.

# etc.<ref name="moloy_de">{{Citation|url=https://www.researchgate.net/profile/Moloy_De/publication/317116429_Second_collection_of_my_hundred_posts_in_the_Facebook_Group_Math's_Believe_It_Or_Not/links/5926e024aca27295a80029f9/Second-collection-of-my-hundred-posts-in-the-Facebook-Group-Maths-Believe-It-Or-Not.pdf|title=MATH’S BELIEVE IT OR NOT|last=De|first=Moloy}}</ref>

# The number formed by its first four digits ''abcd'' is a multiple of 4.

# etc.

For example, EX9876 is a six-digit polydivisible number, but 123456 is not, because 12345 is not a multiple of 5.

==Definition==

There are 71822 polydivisible numbers, and the largest of them is 24-digit 606890346850EX6800E036206464 (this number is the only one 24-digit polydivisible number, and the three 23-digit polydivisible numbers are 3894406094803000760060201X6, 3X9806400220806890509X804X6, 606890346850EX6800E03620646)

Let <math>n</math> be a natural number, and let <math>k = \lfloor \log_{b}{n} \rfloor + 1</math> be the number of digits in the number in base <math>b</math>. <math>n</math> is a '''polydivisible number''' if for all <math>0 \leq i < k</math>,

: <math>\frac{n - (n \bmod b^{k - i - 1})}{b^{k - i - 1}} \equiv 0 \bmod i</math>.

For example, 10801 is a seven-digit polydivisible number in [[base 4]], as

{|class="wikitable"

: <math>\frac{10801 - (10801 \bmod 4^{7 - 0 - 1})}{4^{7 - 0 - 1}} = \frac{10801 - (10801 \bmod 4^{6})}{4^{6}} = \frac{10801 - (10801 \bmod 4096)}{4096} = \frac{10801 - 2609}{4096} = \frac{8192}{4096} = 2 \equiv 0 \bmod 1</math>

|''n''||smallest ''n''-digit polydivisible number||largest ''n''-digit polydivisible number||number of ''n''-digit polydivisible numbers||excepted number of ''n''-digit polydivisible numbers

: <math>\frac{10801 - (10801 \bmod 4^{7 - 1 - 1})}{4^{7 - 1 - 1}} = \frac{10801 - (10801 \bmod 4^{5})}{4^{5}} = \frac{10801 - (10801 \bmod 1024)}{1024} = \frac{10801 - 561}{1024} = \frac{10240}{1024} = 10 \equiv 0 \bmod 2</math>

: <math>\frac{10801 - (10801 \bmod 4^{7 - 2 - 1})}{4^{7 - 2 - 1}} = \frac{10801 - (10801 \bmod 4^{4})}{4^{4}} = \frac{10801 - (10801 \bmod 256)}{256} = \frac{10801 - 49}{256} = \frac{10752}{256} = 42 \equiv 0 \bmod 3</math>

: <math>\frac{10801 - (10801 \bmod 4^{7 - 3 - 1})}{4^{7 - 3 - 1}} = \frac{10801 - (10801 \bmod 4^{3})}{4^{3}} = \frac{10801 - (10801 \bmod 64)}{64} = \frac{10801 - 49}{64} = \frac{10752}{64} = 168 \equiv 0 \bmod 4</math>

: <math>\frac{10801 - (10801 \bmod 4^{7 - 4 - 1})}{4^{7 - 4 - 1}} = \frac{10801 - (10801 \bmod 4^{2})}{4^{2}} = \frac{10801 - (10801 \bmod 16)}{16} = \frac{10801 - 1}{16} = \frac{10800}{16} = 675 \equiv 0 \bmod 5</math>

: <math>\frac{10801 - (10801 \bmod 4^{7 - 5 - 1})}{4^{7 - 5 - 1}} = \frac{10801 - (10801 \bmod 4^{1})}{4^{1}} = \frac{10801 - (10801 \bmod 4)}{4} = \frac{10801 - 1}{4} = \frac{10800}{4} = 2700 \equiv 0 \bmod 6</math>

: <math>\frac{10801 - (10801 \bmod 4^{7 - 6 - 1})}{4^{7 - 6 - 1}} = \frac{10801 - (10801 \bmod 4^{0})}{4^{0}} = \frac{10801 - (10801 \bmod 1)}{1} = \frac{10801 - 0}{1} = \frac{10801}{1} = 10801 \equiv 0 \bmod 7</math>

==Enumeration==

For any given base <math>b</math>, there are only a finite number of polydivisible numbers.

===Maximum polydivisible number===

All numbers are represented in base <math>b</math>, using A−Z to represent digit values 10 to 35.

{| class="wikitable"

|-

! Base <math>b</math>

|1||1||E||E||E

! Maximum polydivisible number

! Number of digits in maximum polydivisible number

|-

| [[base 2|2]] || 10 ||EX||56||56 2

|-

| [[base 3|3]] || 20 0220 || 6

|3||100||EX9||1X0||1X0

|-

| [[base 4|4]] || 222 0301 || 7

|4||1000||EX98||560||560

|-

| [[base 5|5]] || 40220 42200 || 10

|5||10004||EX987||1124||1124.972497249724972497249725

|-

| [[base 10|10]] || 36085 28850 36840 07860 36725<ref name="Parker" /><ref name="Wells">{{Citation|last=Wells|first=David|title=The Penguin Dictionary of Curious and Interesting Numbers|page=197|publisher=Penguin Books|year=1986|via=Google Books|url=https://books.google.com/books?id=kQRPkTkk_VIC&pg=PA197#v=onepage&q&f=false}}</ref><ref name="Lines">{{Citation|last=Lines|first=Malcolm|title=A Number for your Thoughts|chapter=How Do These Series End?|page=90|publisher=Taylor and Francis Group|year=1986|url=https://books.google.com/books?id=Am9og6q_ny4C&pg=PA90#v=onepage&q&f=false}}</ref> || 25<ref name="Parker" /><ref name="Wells"/><ref name="Lines" />

|6||100040||EX9876||2248||2249.72497249724972497249724X

|-

| [[base 12|12]] || 6068 903468 50BA68 00B036 206464 || 28

|-

|}

|7||1000404||EX9876X||392X||3931.0414559E39310414559E3931

===Estimate for <math>F_b(n)</math> and <math>\Sigma(b)</math>===

[[File:Graph of polydivisible number vectorial.svg|right|thumb|400px|Graph of number of <math>n</math>-digit polydivisible numbers in base 10 <math>F_{10}(n)</math> vs estimate of <math>F_{10}(n)</math>]]

Let <math>n</math> be the number of digits. The function <math>F_b(n)</math> determines the number of polydivisible numbers that has <math>n</math> digits in base <math>b</math>, and the function <math>\Sigma(b)</math> is the total number of polydivisible numbers in base <math>b</math>.

If <math>k</math> is a polydivisible number in base <math>b</math> with <math>n - 1</math> digits, then it can be extended to create a polydivisible number with <math>n</math> digits if there is a number between <math>bk</math> and <math>b(k + 1) - 1</math> that is divisible by <math>n</math>. If <math>n</math> is less or equal to <math>b</math>, then it is always possible to extend an <math>n - 1</math> digit polydivisible number to an <math>n</math>-digit polydivisible number in this way, and indeed there may be more than one possible extension. If <math>n</math> is greater than <math>b</math>, it is not always possible to extend a polydivisible number in this way, and as <math>n</math> becomes larger, the chances of being able to extend a given polydivisible number become smaller. On average, each polydivisible number with <math>n - 1</math> digits can be extended to a polydivisible number with <math>n</math> digits in <math>\frac{b}{n}</math> different ways. This leads to the following estimate for <math>F_{b}(n)</math> :

:<math>F_b(n) \approx (b - 1)\frac{b^{n-1}}{n!}.</math>

Summing over all values of n, this estimate suggests that the total number of polydivisible numbers will be approximately

:<math>\Sigma(b) \approx \frac{b - 1}{b}(e^{b}-1)</math>

{| class="wikitable"

|-

! Base <math>b</math>

|8||10004040||EX9876X8||57X4||57X7.6620828XE7X76620828XE7X8

! <math>\Sigma(b)</math>

! Est. of <math>\Sigma(b)</math>

! Percent Error

|-

| [[base 2|2]] || 2 || <math>\frac{1}{2}(e^{2} - 1) \approx 3.1945</math> || 59.7%

|9||100040400||EX9876X83||765X||7662.0828XE7X76620828XE7X7662

|-

| [[base 3|3]] || 15 || <math>\frac{2}{3}(e^{3} - 1) \approx 12.725</math> || -15.1%

|X||1000404008||EX9876X836||8E53||9074.X50X8447E55009X5X9231X27

|-

| [[base 4|4]] || 37 || <math>\frac{3}{4}(e^{4} - 1) \approx 40.199</math> || 8.64%

|E||10004040085||EX9876X836X||987E||9X5X.9231X27328111EX430257584

|-

| [[base 5|5]] || 127 || <math>\frac{4}{5}(e^{5} - 1) \approx 117.93</math> || −7.14%

|10||100040400850||EX9876X836X0||987E||9X5X.9231X27328111EX430257584

|-

| [[base 10|10]] || 20456<ref name="Parker" /> || <math>\frac{9}{10}(e^{10} - 1) \approx 19823</math> || -3.09%

|11||1000404008507||EX9876X836X01||9084||9146.2E3X024X4393X1877497E621

|-

|}

|12||1000404008507X||EX9876X836X014||7922||7990.263353938X116E9147669X53

==Specific bases==

All numbers are represented in base <math>b</math>, using A−Z to represent digit values 10 to 35.

===Base 2===

{| class="wikitable"

|-

! Length ''n''

|13||100040409040463||EX9876X836X0143||6278||6300.2027679X23337E9838530842

! F₂(''n'')

! Est. of F₂(''n'')

! Polydivisible numbers

|-

| 1 || 1 || 1 || 1

|14||100040409X500000||EX9876X004103600||4813||4830.161E7EX478558EX3293E3632

|-

| 2 || 1 || 1 || 10

|15||100046283XX042000||EX982694648006005||3360||3385.89EE920333X2790X9029472E

|-

|}

|16||100046283XX0420000||EX9820483000XX0016||2226||2257.9X7EX14222699207201X309E

===Base 3===

{| class="wikitable"

|-

! Length ''n''

|17||100460000X106000769||EX9820483000XX00167||1465||1487.5744E5333362464200120813

! F₃(''n'')

! Est. of F₃(''n'')

! Polydivisible numbers

|-

| 1 || 2 || 2 || 1, 2

|18||10046008329084680044||EX9820483000XX001674||9E7||X04.5927931E6938762600085256

|-

| 2 || 3 || 3 || 11, 20, 22

|19||1030700092E0X09460249||EX9406683810689490703||581||589.50161X46737326E86X39E667

|-

| 3 || 3 || 3 || 110, 200, 220

|1X||1030700092E0X094602490||EX9080000X70E200006860||311||316.289676903E950397E4655259

|-

| 4 || 3 || 2 || 1100, 2002, 2200

|1E||1030700092E0X0946024900||EX9030683410280000X0944||16X||176.X9981X37EEX7X0E37602X030

|-

| 5 || 2 || 1 || 11002, 20022

|20||1030700092E0X09460249000||EX9030683410280000X09440||90||99.54XX0E19EEE3E05799015016

|-

| 6 || 2 || 1 || 110020, 200220

|21||1030700092E0X09460249000E||EX60562008706X94X67898401||39||48.4623X63915305370481E878E

|-

| 7 || 0 || 0 || <math>\varnothing</math>

|22||109870X09290400016903X0074||EX60562008706X94X678984014||18||22.02X37764261499E678473098

|-

|}

|23||3894406094803000760060201X6||606890346850EX6800E03620646||3||E.6932E481X547585175086843

===Base 4===

{| class="wikitable"

|-

! Length ''n''

|24||606890346850EX6800E036206464||606890346850EX6800E036206464||1||4.E582E8X4291E93741571E537

! F₄(''n'')

! Est. of F₄(''n'')

! Polydivisible numbers

|-

| 1 || 3 || 3 || 1, 2, 3

|25||(none)||(none)||0||2.0739E092EX9E0E5136059352

|-

| 2 || 6 || 6 || 10, 12, 20, 22, 30, 32

|26||(none)||(none)||0||0.9X16451371E2046X14996146

|-

| 3 || 8 || 8 || 102, 120, 123, 201, 222, 300, 303, 321

|27||(none)||(none)||0||0.39887XX698X625782110E7X8

|-

| 4 || 8 || 8 || 1020, 1200, 1230, 2010, 2220, 3000, 3030, 3210

|28||(none)||(none)||0||0.15192E6E679E3E14694XX456

|-

| 5 || 7 || 6 || 10202, 12001, 12303, 20102, 22203, 30002, 32103

|29||(none)||(none)||0||0.0629XX90E190X1X3X12X5E53

|-

| 6 || 4 || 4 || 120012, 123030, 222030, 321030

|2X||(none)||(none)||0||0.0224XXX31996799EE3E72983

|-

| 7 || 1 || 2 || 2220301

|2E||(none)||(none)||0||0.00907X29EE2E2408261E0753

|-

| 8 || 0 || 1 || <math>\varnothing</math>

|-

|30||(none)||(none)||0||0.0030274E3E8E89428X078259

|}

===Base 5===

By above section, no polydivisible numbers exists for ''n''>24

The polydivisible numbers in base 5 are

:1, 2, 3, 4, 11, 13, 20, 22, 24, 31, 33, 40, 42, 44, 110, 113, 132, 201, 204, 220, 223, 242, 311, 314, 330, 333, 402, 421, 424, 440, 443, 1102, 1133, 1322, 2011, 2042, 2200, 2204, 2231, 2420, 2424, 3113, 3140, 3144, 3302, 3333, 4022, 4211, 4242, 4400, 4404, 4431, 11020, 11330, 13220, 20110, 20420, 22000, 22040, 22310, 24200, 24240, 31130, 31400, 31440, 33020, 33330, 40220, 42110, 42420, 44000, 44040, 44310, 110204, 113300, 132204, 201102, 204204, 220000, 220402, 223102, 242000, 242402, 311300, 314000, 314402, 330204, 333300, 402204, 421102, 424204, 440000, 440402, 443102, 1133000, 1322043, 2011021, 2042040, 2204020, 2420003, 2424024, 3113002, 3140000, 3144021, 4022042, 4211020, 4431024, 11330000, 13220431, 20110211, 20420404, 24200031, 31400004, 31440211, 40220422, 42110202, 44310242, 132204314, 201102110, 242000311, 314000044, 402204220, 443102421, 1322043140, 2011021100, 3140000440, 4022042200

The smallest base 5 polydivisible numbers with ''n'' digits are

If ''k'' is a polydivisible number with ''n''-1 digits, then it can be extended to create a polydivisible number with ''n'' digits if there is a number between 10''k'' and 10''k''+E that is divisible by ''n''. If ''n'' is less or equal to 10, then it is always possible to extend an (''n''-1)-digit polydivisible number to an ''n''-digit polydivisible number in this way, and indeed there may be more than one possible extension. If ''n'' is greater than 10, it is not always possible to extend a polydivisible number in this way, and as ''n'' becomes larger, the chances of being able to extend a given polydivisible number become smaller (e.g. for ''n''=14, the chance is 3/4 or 90%, and for ''n''=16, the chance is 2/3 or 80%, and for ''n''=20, the chance is 1/2 or 60%). On average, each polydivisible number with ''n''-1 digits can be extended to a polydivisible number with ''n'' digits in 10/''n'' different ways. This leads to the following estimate for ''F(n)'' :

:1, 11, 110, 1102, 11020, 110204, 1133000, 11330000, 132204314, 1322043140, 0, 0, 0...

The largest base 5 polydivisible numbers with ''n'' digits are

''F''(''n'') ≈ (E*10^(''n''-1))/(''n''!)

:4, 44, 443, 4431, 44310, 443102, 4431024, 44310242, 443102421, 4022042200, 0, 0, 0...

The number of base 5 polydivisible numbers with ''n'' digits are

Summing over all values of ''n'', this estimate suggests that the total number of polydivisible numbers will be approximately

:4, 10, 17, 21, 21, 21, 13, 10, 6, 4, 0, 0, 0...

{| class="wikitable"

E*(''e''^10-1)/10 = 72406.E857361390E1XE713X815171...

|-

! Length ''n''

! F₅(''n'')

! Est. of F₅(''n'')

|-

| 1 || 4 || 4

|-

| 2 || 10 || 10

|-

| 3 || 17 || 17

|-

| 4 || 21 || 21

|-

| 5 || 21 || 21

|-

| 6 || 21 || 17

|-

| 7 || 13 || 12

|-

| 8 || 10 || 8

|-

| 9 || 6 || 4

|-

| 10 || 4 || 2

|-

|}

===Base 10===

where ''e'' = 2.8752360698219EX71971009E... is the base of natural logarithm.

The polydivisible numbers in base 10 are

:1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 102, 105, 108, 120, 123, 126, 129, 141, 144, 147, 162, 165, 168, 180, 183, 186, 189, ... {{OEIS|id=A144688}}

The smallest base 10 polydivisible numbers with ''n'' digits are

There are about E*10^(''n''-1)/''n''! ''n''-digit polydivisible numbers.

:1, 10, 102, 1020, 10200, 102000, 1020005, 10200056, 102000564, 1020005640, 10200056405, 102006162060, 1020061620604, 10200616206046, 102006162060465, 1020061620604656, 10200616206046568, 108054801036000018, 1080548010360000180, 10805480103600001800, ... {{OEIS|id=A214437}}

The largest base 10 polydivisible numbers with ''n'' digits are

:9, 98, 987, 9876, 98765, 987654, 9876545, 98765456, 987654564, 9876545640, 98765456405, 987606963096, 9876069630960, 98760696309604, 987606963096045, 9876062430364208, 98485872309636009, 984450645096105672, 9812523240364656789, 96685896604836004260, ... {{OEIS|id=A225608}}

The number of base 10 polydivisible numbers with ''n'' digits are

:9, 45, 150, 375, 750, 1200, 1713, 2227, 2492, 2492, 2225, 2041, 1575, 1132, 770, 571, 335, 180, 90, 44, 18, 12, 6, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... {{OEIS|id=A143671}}

{| class="wikitable" style="float:left; margin-right:1em"

|-

! Length ''n''

! F₁₀(''n'')<ref name="A143671">{{OEIS|id=A143671}}</ref>

! Est. of F₁₀(''n'')

|-

| 1

| 9

| 9

|-

| 2

| 45

| 45

|-

| 3

| 150

| 150

|-

| 4

| 375

| 375

|-

| 5

| 750

| 750

|-

| 6

| 1200

| 1250

|-

| 7

| 1713

| 1786

|-

| 8

| 2227

| 2232

|-

| 9

| 2492

| 2480

|-

| 10

| 2492

| 2480

|}

{| class="wikitable" style="float:left; margin-right:1em"

|-

! Length ''n''

! F₁₀(''n'') <ref name="A143671" />

! Est. of F₁₀(''n'')

|-

| 11

| 2225

| 2255

|-

| 12

| 2041

| 1879

|-

| 13

| 1575

| 1445

|-

| 14

| 1132

| 1032

|-

| 15

| 770

| 688

|-

| 16

| 571

| 430

|-

| 17

| 335

| 253

|-

| 18

| 180

| 141

|-

| 19

| 90

| 74

|-

| 20

| 44

| 37

|}

{| class="wikitable" style="float:left; margin-right:1em"

|-

! Length ''n''

! F₁₀(''n'') <ref name="A143671" />

! Est. of F₁₀(''n'')

|-

| 21

| 18

| 17

|-

| 22

| 12

| 8

|-

| 23

| 6

| 3

|-

| 24

| 3

| 1

|-

| 25

| 1

| 1

|}{{clear left}}

==Programming example==

There are no E-digit polydivisible numbers using all the digits 1 to E exactly once. (hence there are also no 10-digit polydivisible numbers using all the digits 0 to E exactly once, since if a number with digits ''abcdefghijkl'' is a 10-digit polydivisible number using all the digits 0 to E exactly once, then {''a'', ''b'', ''c'', ''d'', ''e'', ''f'', ''g'', ''h'', ''i'', ''j'', ''k'', ''l''} = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, X, E}, and then ''abcdefghijkl'' is divisible by 10, thus we have ''l'' = 0 (by [[divisibility rule]] of 10), and {''a'', ''b'', ''c'', ''d'', ''e'', ''f'', ''g'', ''h'', ''i'', ''j'', ''k''} = {1, 2, 3, 4, 5, 6, 7, 8, 9, X, E}, thus a number with digits ''abcdefghijk'' is an E-digit polydivisible numbers using all the digits 1 to E exactly once).

The example below searches for polydivisible numbers in [[Python (programming language)|Python]].

<syntaxhighlight lang="python">

def find_polydivisible(base: int) -> List[int]:

"""Find polydivisible number."""

numbers = []

previous = []

for i in range(1, base):

previous.append(i)

new = []

digits = 2

while not previous == []:

numbers.append(previous)

for i in range(0, len(previous)):

for j in range(0, base):

number = previous[i] * base + j

if number % digits == 0:

new.append(number)

previous = new

new = []

digits = digits + 1

return numbers

</syntaxhighlight>

==Related problems==

Proof: if a number with digits ''abcdefghijk'' is an E-digit polydivisible numbers using all the digits 1 to E exactly once, then {''a'', ''b'', ''c'', ''d'', ''e'', ''f'', ''g'', ''h'', ''i'', ''j'', ''k''} = {1, 2, 3, 4, 5, 6, 7, 8, 9, X, E}, and we have:

Polydivisible numbers represent a generalization of the following well-known<ref name="Parker">{{Citation|last=Parker|first=Matt|title=Things to Make and Do in the Fourth Dimension|chapter=Can you digit?|pages=7-8|year=2014|publisher=Particular Books|via=Google Books|url=https://books.google.com/books?id=veIxBQAAQBAJ&pg=PA8#v=onepage&q&f=false}}</ref> problem in [[recreational mathematics]] :

: ''Arrange the digits 1 to 9 in order so that the first two digits form a multiple of 2, the first three digits form a multiple of 3, the first four digits form a multiple of 4 etc. and finally the entire number is a multiple of 9.''

''f'' = 6 (since ''abcdef'' is divisible by 6) (by [[divisibility rule]] of 6)

The solution to the problem is a nine-digit polydivisible number with the additional condition that it contains the digits 1 to 9 exactly once each. There are 2,492 nine-digit polydivisible numbers, but the only one that satisfies the additional condition is

{''d'', ''h''} = {4, 8} (since ''abcd'' is divisible by 4 and ''abcdefgh'' is divisible by 8 (thus by 4)) (by [[divisibility rule]] of 4)

:'''381 654 729'''<ref name="Lanier">{{Citation|url=http://jwilson.coe.uga.edu/emt725/Class/Lanier/Nine.Digit/nine.html|first=Susie|last=Lanier|title=Nine Digit Number}}</ref>

{''c'', ''i''} = {3, 9} (since ''abc'' is divisible by 3 and ''abcdefghi'' is divisible by 9 (thus by 3)) (by [[divisibility rule]] of 3)

Other problems involving polydivisible numbers include:

{''b'', ''j''} = {2, X} (since ''ab'' is divisible by 2 and ''abcdefghij'' is divisible by X (thus by 2)) (by [[divisibility rule]] of 2)

* Finding polydivisible numbers with additional restrictions on the digits - for example, the longest polydivisible number that only uses even digits is

thus, we have {''a'', ''e'', ''g'', ''k''} = {1, 5, 7, E}

:'''480 006 882 084 660 840 40'''

Since ''abcdefgh'' is divisible by 8, thus ''gh'' is divisible by 8 (by [[divisibility rule]] of 8), and since {''a'', ''e'', ''g'', ''k''} = {1, 5, 7, E}, thus ''g'' is odd, and ''h'' must be 4 (if ''h'' = 8 and ''g'' is odd, then ''gh'' is not divisible by 8), and since ''abcdefghi'' is divisible by 9, thus ''hi'' is divisible by 9 (by [[divisibility rule]] of 9), however, ''h'' = 4 and ''i'' is either 3 or 9, but neither 43 nor 49 is divisible by 9, a contradiction!

* Finding [[palindromic number|palindromic]] polydivisible numbers - for example, the longest palindromic polydivisible number is

If we do not require the number formed by its first 8 digits divisible by 8, then there are 2 solutions: 1X98265E347 and 7298X65E341 (neither satisfies that the number formed by its first 8 digits is divisible by 4, even neither satisfies that the number formed by its first 8 digits is divisible by 2).

:'''300 006 000 03'''

If we do not require the number formed by its first 9 digits divisible by 9, then there are 4 solutions: 1X38E694725, 7X981634E25, 7X98E654321, and EX987634125 (only 7X98E654321 satisfies that the number formed by its first 9 digits is divisible by 3).

* A common, trivial extension of the aforementioned example is to arrange the digits 0 to 9 to make a 10 digit number in the same way, the result is 3816547290. This is a [[pandigital]] polydivisible number.

Except this case of dozenal (i.e. base 10), such number exists in all even bases < 14 (and does not exist in any odd base since the number formed by the first (base−1) digits cannot be divisible by (base−1)), however, such number also does not exist in any even base 14 ≤ ''b'' ≤ 48.

==References==

Although there are no 10-digit polydivisible number with distinct digits, the only one E-digit polydivisible number with distinct digits is 7X981054623, and there are 5 X-digit polydivisible number with distinct digits: 7X98105462, E09456283X, E49016283X, EX94502836, and EX98705462

{{reflist}}

==External links==

== Related problems ==

* [https://www.youtube.com/watch?v=gaVMrqzb91w YouTube - a pandigital number that is also polydivisible]

{{Classes of natural numbers}}

* Finding polydivisible numbers with additional restrictions on the digits - for example, the longest polydivisible numbers that only uses even digits are 260000800600260046X4 and 2X646640024000680044 (both has 18 digits)

{{Divisor classes}}

[[Category:Base-dependent integer sequences]]

* Finding palindromic polydivisible numbers - for example, the longest palindromic polydivisible numbers are 42643634624 and 8X3840483X8 (both has E digits)

[[Category:Modular arithmetic]]