Polydivisible number: Difference between revisions

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

# 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>

==Definition==

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

: <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>

: <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==