In mathematics a polydivisible number (or magic number) is a number in a given number base with digits abcde... 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.
Definition Let n {\displaystyle n} be a positive integer, and let k = ⌊ log b n ⌋ + 1 {\displaystyle k=\lfloor \log _{b}{n}\rfloor +1} be the number of digits in n written in base b. The number n is a polydivisible number if for all 1 ≤ i ≤ k {\displaystyle 1\leq i\leq k} ,
⌊ n b k − i ⌋ ≡ 0 ( mod i ) {\displaystyle \left\lfloor {\frac {n}{b^{k-i}}}\right\rfloor \equiv 0{\pmod {i}}} . Example For example, 10801 is a seven-digit polydivisible number in base 4, as
⌊ 10801 4 7 − 1 ⌋ = ⌊ 10801 4096 ⌋ = 2 ≡ 0 ( mod 1 ) , {\displaystyle \left\lfloor {\frac {10801}{4^{7-1}}}\right\rfloor =\left\lfloor {\frac {10801}{4096}}\right\rfloor =2\equiv 0{\pmod {1}},}
⌊ 10801 4 7 − 2 ⌋ = ⌊ 10801 1024 ⌋ = 10 ≡ 0 ( mod 2 ) , {\displaystyle \left\lfloor {\frac {10801}{4^{7-2}}}\right\rfloor =\left\lfloor {\frac {10801}{1024}}\right\rfloor =10\equiv 0{\pmod {2}},}
⌊ 10801 4 7 − 3 ⌋ = ⌊ 10801 256 ⌋ = 42 ≡ 0 ( mod 3 ) , {\displaystyle \left\lfloor {\frac {10801}{4^{7-3}}}\right\rfloor =\left\lfloor {\frac {10801}{256}}\right\rfloor =42\equiv 0{\pmod {3}},}
⌊ 10801 4 7 − 4 ⌋ = ⌊ 10801 64 ⌋ = 168 ≡ 0 ( mod 4 ) , {\displaystyle \left\lfloor {\frac {10801}{4^{7-4}}}\right\rfloor =\left\lfloor {\frac {10801}{64}}\right\rfloor =168\equiv 0{\pmod {4}},}
⌊ 10801 4 7 − 5 ⌋ = ⌊ 10801 16 ⌋ = 675 ≡ 0 ( mod 5 ) , {\displaystyle \left\lfloor {\frac {10801}{4^{7-5}}}\right\rfloor =\left\lfloor {\frac {10801}{16}}\right\rfloor =675\equiv 0{\pmod {5}},}
⌊ 10801 4 7 − 6 ⌋ = ⌊ 10801 4 ⌋ = 2700 ≡ 0 ( mod 6 ) , {\displaystyle \left\lfloor {\frac {10801}{4^{7-6}}}\right\rfloor =\left\lfloor {\frac {10801}{4}}\right\rfloor =2700\equiv 0{\pmod {6}},}
⌊ 10801 4 7 − 7 ⌋ = ⌊ 10801 1 ⌋ = 10801 ≡ 0 ( mod 7 ) . {\displaystyle \left\lfloor {\frac {10801}{4^{7-7}}}\right\rfloor =\left\lfloor {\frac {10801}{1}}\right\rfloor =10801\equiv 0{\pmod {7}}.}
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