Pandigital Properties of 123,456,789

Theorem

$123 \, 456 \, 789$ has the following properties:

It is pandigital, and remains so when multiplied by $2$, $4$, $5$, $7$ and $8$:

 $\displaystyle 123 \, 456 \, 789 \times 1$ $=$ $\displaystyle 123 \, 456 \, 789$ $\displaystyle 123 \, 456 \, 789 \times 2$ $=$ $\displaystyle 246 \, 913 \, 578$ $\displaystyle 123 \, 456 \, 789 \times 3$ $=$ $\displaystyle 370 \, 370 \, 367$ $\displaystyle 123 \, 456 \, 789 \times 4$ $=$ $\displaystyle 493 \, 827 \, 156$ $\displaystyle 123 \, 456 \, 789 \times 5$ $=$ $\displaystyle 617 \, 283 \, 945$ $\displaystyle 123 \, 456 \, 789 \times 6$ $=$ $\displaystyle 740 \, 740 \, 734$ $\displaystyle 123 \, 456 \, 789 \times 7$ $=$ $\displaystyle 864 \, 197 \, 523$ $\displaystyle 123 \, 456 \, 789 \times 8$ $=$ $\displaystyle 987 \, 654 \, 312$ $\displaystyle 123 \, 456 \, 789 \times 9$ $=$ $\displaystyle 1 \, 111 \, 111 \, 101$