Permuted Multiples

It can be seen that the number, $125874$ , and its double, $251748$ , contain exactly the same digits, but in a different order.

Find the smallest positive integer, $x$ , such that $2 x$ , $3 x$ , $4 x$ , $5 x$ , and $6 x$ , contain the same digits.

Write-up coming later

The complete problem is available here. An approach, code, and answer will be added later.