Definition:Polydivisible Number

Definition

Definition 1

A polydivisible number is a positive integer $N$ of which, for all $n$ up to the number of digits of $N$, the first $n$ digits form an integer which is divisible by $n$.

Definition 2

All integers from $1$ to $9$ are defined as being polydivisible.

A positive integer $N$ such that $N \ge 10$ is polydivisible if and only if:

$(1): \quad$ It is divisible by the number of its digits

$(2): \quad$ The integer which remains when its last digit is deleted is also polydivisible.

Sequence of Polydivisible Numbers

The sequence of polydivisible numbers begins:

$1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, \ldots$

Examples

$381 \, 654 \, 729$ is Polydivisible

The integer $381 \, 654 \, 729$ is the only polydivisible number which is penholodigital.

$3 \, 608 \, 528 \, 850 \, 368 \, 400 \, 786 \, 036 \, 725$ is Polydivisible

The largest polydivisible number has $25$ digits:

$3 \, 608 \, 528 \, 850 \, 368 \, 400 \, 786 \, 036 \, 725$

Also known as

Some sources call such integers magic numbers

Also see

• Equivalence of Definitions of Polydivisible Number