Problem 43

Sub-string Divisibility

The number, $1406357289$ , is a $0$ to $9$ pandigital number because it is made up of each of the digits $0$ to $9$ in some order, but it also has a rather interesting sub-string divisibility property.

Let $d_{1}$ be the $1$ ^st digit, $d_{2}$ be the $2$ ^nd digit, and so on. In this way, we note the following:

$d_{2} d_{3} d_{4} = 406$ is divisible by $2$
$d_{3} d_{4} d_{5} = 063$ is divisible by $3$
$d_{4} d_{5} d_{6} = 635$ is divisible by $5$
$d_{5} d_{6} d_{7} = 357$ is divisible by $7$
$d_{6} d_{7} d_{8} = 572$ is divisible by $11$
$d_{7} d_{8} d_{9} = 728$ is divisible by $13$
$d_{8} d_{9} d_{10} = 289$ is divisible by $17$

Find the sum of all $0$ to $9$ pandigital numbers with this property.

子串的可整除性

$1406357289$ 是一个 $0$ 至 $9$ 全数字数，因为它由 $0$ 到 $9$ 这十个数字排列而成；但除此之外，它还有一个有趣的性质：子串的可整除性。

记 $d_{1}$ 是它的第 $1$ 个数字， $d_{2}$ 是第 $2$ 个数字，依此类推，注意到：

$d_{2} d_{3} d_{4} = 406$ 能被 $2$ 整除
$d_{3} d_{4} d_{5} = 063$ 能被 $3$ 整除
$d_{4} d_{5} d_{6} = 635$ 能被 $5$ 整除
$d_{5} d_{6} d_{7} = 357$ 能被 $7$ 整除
$d_{6} d_{7} d_{8} = 572$ 能被 $11$ 整除
$d_{7} d_{8} d_{9} = 728$ 能被 $13$ 整除
$d_{8} d_{9} d_{10} = 289$ 能被 $17$ 整除

找出所有满足同样性质的 $0$ 至 $9$ 全数字数，并求它们的和。