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| In [[mathematics]] a '''polydivisible number''' is a [[natural number|number]] with [[numerical digit|digits]] ''abcde...'' that has the following properties :
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| # Its first digit ''a'' is not 0.
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| # The number formed by its first two digits ''ab'' is a multiple of 2.
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| # The number formed by its first three digits ''abc'' is a multiple of 3.
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| # The number formed by its first four digits ''abcd'' is a multiple of 4.
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| # etc.
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| For example, 345654 is a six-digit polydivisible number, but 123456 is not, because 1234 is not a multiple of 4. Polydivisible numbers can be defined in any [[radix|base]] - however, the numbers in this article are all in base 10, so permitted digits are 0 to 9.
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| The smallest base 10 polydivisible numbers with 1,2,3,4... etc. digits are
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| [[1 (number)|1]], [[10 (number)|10]], [[102 (number)|102]], 1020, 10200, 102000, 1020005, 10200056, 102000564, 1020005640 {{OEIS|id=A078282}}
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| ==Background==
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| Polydivisible numbers are a generalisation of the following well-known problem in [[recreational mathematics]] :
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| : ''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.''
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| 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 | |
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| :'''381654729'''
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| ==How many polydivisible numbers are there?==
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| 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''+9 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.
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| 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 of the number of ''n''-digit polydivisible numbers, which we will denote by ''F(n)'' :
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| :<math>F(n) \approx \frac{9 \times 10^{n-1}}{n!}</math>
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| Summing over all values of n, this estimate suggests that the total number of polydivisible numbers will be approximately
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| :<math>\frac{9(e^{10}-1)}{10}\approx 19823</math>
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| In fact, this underestimates the actual number of polydivisible numbers by about 3%.
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| ==Counting polydivisible numbers==
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| We can find the actual values of ''F(n)'' by counting the number of polydivisible numbers with a given length :
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| [[Image:Graph of polydivisible number vectorial.svg|right|400px]]
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| <table border="1" cellpadding="2">
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| <tr><th>Length ''n''<th>F(''n'')<th>Estimate of F(''n'')<th><th>Length ''n''<th>F(''n'')<th>Estimate of F(''n'')<th><th>Length ''n''<th>F(''n'')<th>Estimate of F(''n'')
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| <tr><td>1<td>9<td>9<td><td>11<td>2225<td>2255<td><td>21<td>18<td>17
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| <tr><td>2<td>45<td>45<td><td>12<td>2041<td>1879<td><td>22<td>12<td>8
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| <tr><td>3<td>150<td>150<td><td>13<td>1575<td>1445<td><td>23<td>6<td>3
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| <tr><td>4<td>375<td>375<td><td>14<td>1132<td>1032<td><td>24<td>3<td>1
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| <tr><td>5<td>750<td>750<td><td>15<td>770<td>688<td><td>25<td>1<td>1
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| <tr><td>6<td>1200<td>1250<td><td>16<td>571<td>430<td>
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| <tr><td>7<td>1713<td>1786<td><td>17<td>335<td>253<td>
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| <tr><td>8<td>2227<td>2232<td><td>18<td>180<td>141<td>
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| <tr><td>9<td>2492<td>2480<td><td>19<td>90<td>74<td>
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| <tr><td>10<td>2492<td>2480<td><td>20<td>44<td>37<td>
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| </table>
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| There are 20,456 polydivisible numbers altogether, and the longest polydivisible number, which has 25 digits, is :
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| :'''360 852 885 036 840 078 603 672 5'''
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| ==Related problems==
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| Other problems involving polydivisible numbers include :
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| * Finding polydivisible numbers with additional restrictions on the digits - for example, the longest polydivisible number that only uses even digits is
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| :'''480 006 882 084 660 840 40'''
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| * Finding [[palindromic number|palindromic]] polydivisible numbers - for example, the longest palindromic polydivisible number is
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| :'''300 006 000 03'''
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| * Enumerating polydivisible numbers in other bases.
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| ==External links==
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| * [http://jwilson.coe.uga.edu/emt725/Class/Lanier/Nine.Digit/nine.html The nine-digit problem and its solution]
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| * [http://www.filmshuren.nl/randomstuff/polydivisiblenumbers.txt A list of all 20,456 polydivisible numbers]
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| {{Classes of natural numbers}}
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| [[Category:Base-dependent integer sequences]]
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Teacher of the Reading Impaired Darell from The Pas, spends time with pursuits which include genealogy, new launch property singapore and trekkie. Completed a cruiseship experience that consisted of passing by Kunta Kinteh Island and Related Sites.
Look into my blog; http://www.itexamtest.us/Groups-2/singapore-condo-for-sale-apartment-new-launches-One-balmoral-city-resort-dleedon-Sky-habitat-goodwood/