# Pandigital Properties of 987,654,321

## Theorem

$987 \, 654 \, 321$ has the following properties:

It is pandigital, and remains so when multiplied by $1$, $2$, $4$, $5$, $7$ and $8$:

 $\displaystyle 987 \, 654 \, 321 \times 1$ $=$ $\displaystyle 987 \, 654 \, 321$ $\displaystyle 987 \, 654 \, 321 \times 2$ $=$ $\displaystyle 1 \, 975 \, 308 \, 642$ $\displaystyle 987 \, 654 \, 321 \times 3$ $=$ $\displaystyle 2 \, 962 \, 962 \, 963$ $\displaystyle 987 \, 654 \, 321 \times 4$ $=$ $\displaystyle 3 \, 950 \, 617 \, 284$ $\displaystyle 987 \, 654 \, 321 \times 5$ $=$ $\displaystyle 4 \, 938 \, 271 \, 605$ $\displaystyle 987 \, 654 \, 321 \times 6$ $=$ $\displaystyle 5 \, 925 \, 925 \, 925$ $\displaystyle 987 \, 654 \, 321 \times 7$ $=$ $\displaystyle 6 \, 975 \, 308 \, 642$ $\displaystyle 987 \, 654 \, 321 \times 8$ $=$ $\displaystyle 7 \, 901 \, 234 \, 568$ $\displaystyle 987 \, 654 \, 321 \times 9$ $=$ $\displaystyle 8 \, 888 \, 888 \, 889$

Also:

$987 \, 654 \, 321 - 123 \, 456 \, 789 = 864 \, 197 \, 532$

which is also pandigital.