In mathematics a polydivisible number (or magic number) is a number with digits abcdef... that has the following properties :
- Its first digit a is not 0.
- The number formed by its first two digits ab is a multiple of 2.
- The number formed by its first three digits abc is a multiple of 3.
- 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. Polydivisible numbers can be defined in any base - however, the numbers in this article are all in base 10, so permitted digits are 0 to E.
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)
n | smallest n-digit polydivisible number | largest n-digit polydivisible number | number of n-digit polydivisible numbers | excepted number of n-digit polydivisible numbers |
1 | 1 | E | E | E |
2 | 10 | EX | 56 | 56 |
3 | 100 | EX9 | 1X0 | 1X0 |
4 | 1000 | EX98 | 560 | 560 |
5 | 10004 | EX987 | 1124 | 1124.972497249724972497249725 |
6 | 100040 | EX9876 | 2248 | 2249.72497249724972497249724X |
7 | 1000404 | EX9876X | 392X | 3931.0414559E39310414559E3931 |
8 | 10004040 | EX9876X8 | 57X4 | 57X7.6620828XE7X76620828XE7X8 |
9 | 100040400 | EX9876X83 | 765X | 7662.0828XE7X76620828XE7X7662 |
X | 1000404008 | EX9876X836 | 8E53 | 9074.X50X8447E55009X5X9231X27 |
E | 10004040085 | EX9876X836X | 987E | 9X5X.9231X27328111EX430257584 |
10 | 100040400850 | EX9876X836X0 | 987E | 9X5X.9231X27328111EX430257584 |
11 | 1000404008507 | EX9876X836X01 | 9084 | 9146.2E3X024X4393X1877497E621 |
12 | 1000404008507X | EX9876X836X014 | 7922 | 7990.263353938X116E9147669X53 |
13 | 100040409040463 | EX9876X836X0143 | 6278 | 6300.2027679X23337E9838530842 |
14 | 100040409X500000 | EX9876X004103600 | 4813 | 4830.161E7EX478558EX3293E3632 |
15 | 100046283XX042000 | EX982694648006005 | 3360 | 3385.89EE920333X2790X9029472E |
16 | 100046283XX0420000 | EX9820483000XX0016 | 2226 | 2257.9X7EX14222699207201X309E |
17 | 100460000X106000769 | EX9820483000XX00167 | 1465 | 1487.5744E5333362464200120813 |
18 | 10046008329084680044 | EX9820483000XX001674 | 9E7 | X04.5927931E6938762600085256 |
19 | 1030700092E0X09460249 | EX9406683810689490703 | 581 | 589.50161X46737326E86X39E667 |
1X | 1030700092E0X094602490 | EX9080000X70E200006860 | 311 | 316.289676903E950397E4655259 |
1E | 1030700092E0X0946024900 | EX9030683410280000X0944 | 16X | 176.X9981X37EEX7X0E37602X030 |
20 | 1030700092E0X09460249000 | EX9030683410280000X09440 | 90 | 99.54XX0E19EEE3E05799015016 |
21 | 1030700092E0X09460249000E | EX60562008706X94X67898401 | 39 | 48.4623X63915305370481E878E |
22 | 109870X09290400016903X0074 | EX60562008706X94X678984014 | 18 | 22.02X37764261499E678473098 |
23 | 3894406094803000760060201X6 | 606890346850EX6800E03620646 | 3 | E.6932E481X547585175086843 |
24 | 606890346850EX6800E036206464 | 606890346850EX6800E036206464 | 1 | 4.E582E8X4291E93741571E537 |
25 | (none) | (none) | 0 | 2.0739E092EX9E0E5136059352 |
26 | (none) | (none) | 0 | 0.9X16451371E2046X14996146 |
27 | (none) | (none) | 0 | 0.39887XX698X625782110E7X8 |
28 | (none) | (none) | 0 | 0.15192E6E679E3E14694XX456 |
29 | (none) | (none) | 0 | 0.0629XX90E190X1X3X12X5E53 |
2X | (none) | (none) | 0 | 0.0224XXX31996799EE3E72983 |
2E | (none) | (none) | 0 | 0.00907X29EE2E2408261E0753 |
30 | (none) | (none) | 0 | 0.0030274E3E8E89428X078259 |
By above section, no polydivisible numbers exists for n>24
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 10k and 10k+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) :
F(n) ≈ (E*10^(n-1))/(n!)
Summing over all values of n, this estimate suggests that the total number of polydivisible numbers will be approximately
E*(e^10-1)/10 = 72406.E857361390E1XE713X815171...
where e = 2.8752360698219EX71971009E... is the base of natural logarithm.
There are about E*10^(n-1)/n! n-digit polydivisible numbers.
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).
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:
f = 6 (since abcdef is divisible by 6) (by divisibility rule of 6)
{d, h} = {4, 8} (since abcd is divisible by 4 and abcdefgh is divisible by 8 (thus by 4)) (by divisibility rule of 4)
{c, i} = {3, 9} (since abc is divisible by 3 and abcdefghi is divisible by 9 (thus by 3)) (by divisibility rule of 3)
{b, j} = {2, X} (since ab is divisible by 2 and abcdefghij is divisible by X (thus by 2)) (by divisibility rule of 2)
thus, we have {a, e, g, k} = {1, 5, 7, E}
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!
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)
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)
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
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
Related problems[]
- 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)
- Finding palindromic polydivisible numbers - for example, the longest palindromic polydivisible numbers are 42643634624 and 8X3840483X8 (both has E digits)